Answers
ANSWER
See the explanation for the proof.
EXPLANATION
First we have to write the secant and the cotangent in terms of sine and cosine:
\(\sec x=\frac{1}{\cos x}\)\(\cot x=\frac{\cos x}{\sin x}\)Replace into the equation,
\(-2\mleft(\frac{\cos x}{\sin x}\mright)^2=\frac{1}{\frac{1}{\cos x}+1}-\frac{1}{\frac{1}{\cos x}-1}\)On the left side distribute the square and on the right side do the addition and subtraction in the denominators,
\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{1}{\frac{1+\cos x}{\cos x}}-\frac{1}{\frac{1-\cos x}{\cos x}}\)1 over a fraction is equivalent to the reciprocal of the fraction,
\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{\cos x}{1+\cos x}-\frac{\cos x}{1-\cos x}\)Now do the subtraction on the right side,
\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{\cos x(1-\cos x)-\cos x(1+\cos x)}{(1+\cos x)(1-\cos x)}\)Note that the denominator is a difference of two squares,
\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{\cos x(1-\cos x)-\cos x(1+\cos x)}{1-\cos ^2x}\)Apply the distributive property on the two terms of the numerator,
\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{\cos x-\cos ^2x-\cos x-\cos ^2x}{1-\cos ^2x}\)\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{-\cos ^2x-\cos ^2x}{1-\cos ^2x}\)\(-2\frac{\cos^2x}{\sin^2x}^{}=\frac{-2\cos ^2x}{1-\cos ^2x}\)Now, for the denominator remember the trigonometric identity
\(\sin ^2\alpha+\cos ^2\alpha=1\)If we solve it for sin²α
\(\sin ^2\alpha=1-\cos ^2\alpha\)Therefore, the expression we have in the denominator of the right side of the equation is equivalent to sin²x,
\(-2\frac{\cos^2x}{\sin^2x}^{}=-2\frac{\cos ^2x}{\sin ^2x}\)Now we have on both sides,
\(-2\cot ^2x=-2\cot ^2x\)Hence, we have proven that the two expressions are equivalent.
Related Questions
(02.02 MC)
If trapezoid ABCD was reflected over the y-axis, reflected over the x-axis, and rotated 180°, where would point A′′′ lie?
Trapezoid formed by ordered pairs A at negative 4, 1, B at negative 3, 2, C at negative 1, 2, D at 0, 1.
(1, −1)
(−4, 1)
(1, 1)
(−4, −1)
Answers
The location of point A''' after the three transformations would be (-4, 1).
To determine the location of point A''', we need to apply the three transformations (reflection over the y-axis, reflection over the x-axis, and rotation of 180°) to point A.
When a point is reflected over the y-axis, the x-coordinate is negated while the y-coordinate remains the same.
So, the reflection of point A (-4, 1) over the y-axis would be (4, 1).
When a point is reflected over the x-axis, the y-coordinate is negated while the x-coordinate remains the same. So, the reflection of point (4, 1) over the x-axis would be (4, -1).
When a point is rotated 180°, the x-coordinate and y-coordinate are both negated. So, the rotation of point (4, -1) by 180° would be (-4, 1).
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A max subscription for Twitch, a streaming service, costs a flat fee of $25. You can then also buy small subscriptions for others for $5 per subscription. Johnny is trying to budget the amount of money he wants to spend.
Let y represent the total amount of money spent, and x represent the number of small subscriptions he purchases.
Answers
Answer:
y = 25 + 5x
Step-by-step explanation:
Since the cost of Twitch is $25 no matter what, that will be our y-intercept (essentially when no other subscriptions are bought). Since with every purchase x we make, we spend an additional $5, the slope of the line (x coefficient) will be 5.
Does anyone know how to solve this?
Answers
9514 1404 393
Answer:
zeros: {-2, -1, 7}P(x) = (x +2)(x +1)(x -7)Step-by-step explanation:
Descarte's rule of signs tells you there is 1 positive real root. The rational root theorem tells you any real roots are factors of ±14, so will be ±1, ±2, ±7, or ±14.
We note that the sum of all coefficients is negative, so +1 is not a root.
We note that the sum of coefficients of odd-degree terms is 1-19 = -18, and the sum of coefficients of even-dgree terms is -4-14 = -18. These are the same value, indicating that -1 is a root. The corresponding factor is (x +1).
The result of dividing the factor (x+1) from the polynomial using synthetic division is shown in the first attachment. The coefficients found mean the quotient is ...
Q(x) = x^2 -5x -14
Factoring this using any method you like gives ...
Q(x) = (x +2)(x -7)
This means the full set of zeros is {-2, -1, +7} and the factorization is ...
P(x) = (x +2)(x +1)(x -7)
_____
Additional comment
My personal favorite method of finding the zeros is to let a graphing calculator show them to me. That result is shown in the second attachment.
For each pair of functions f and g below, find f(g(x)) and g (f(x)).
Then, determine whether fand g are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all x in the domain of the composition.
You do not have to indicate the domain.)
f(x) = x+4
g (x) = x+4
Answers
The Function f(g(x)) = g(f(x)) = x + 8.
The functions are: f(x) = x + 4 and g(x) = x + 4. We can find f(g(x)) by substituting g(x) in place of x in f(x).
f(g(x)) = f(x + 4) = (x + 4) + 4 = x + 8
Similarly, we can find g(f(x)) by substituting f(x) in place of x in g(x).g(f(x)) = g(x + 4) = (x + 4) + 4 = x + 8
Thus, we can see that f(g(x)) and g(f(x)) are equal to each other,
which is x + 8.
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Find the slope of the line that passes through each pair of points. Express as a fraction in simplest form. (-6, -2), (-1.5, 5.5)
Answers
Answer:
m = slope= 5/3
Step-by-step explanation:
\((-6, -2) = (x_1,y_1) \\ (-1.5, 5.5) (x_2 ,y_2) \\ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 -x _1} \)
\( \frac{y - ( - 2)}{x - ( - 6)} = \frac{5.5 - ( -2 )}{ - 1.5 - ( - 6)} \\ \frac{y + 2}{x + 6} = \frac{5.5 + 2}{ - 1.5 + 6} \)
\( \frac{y + 2}{x + 6} = \frac{7.5}{4.5} \\ 4.5(y + 2) = 7.5(x + 6)\)
\(4.5y + 9 = 7.5x + 45 \\ 4.5y = 7.5x + 45 - 9 \\ 4.5y = 7.5x + 36\)
Divide through by 4.5
\( \frac{4.5y}{4.5} = \frac{7.5x}{4.5} + \frac{36}{4.5} \\ y = \frac{5}{3} x + 8 \\ \)
the dimension of the confirms room or 7.2 M * 11 M what is the area of the conference room
Answers
Step-by-step explanation:
Just multiply the two dimensions to get m^2 area
7.2 m * 11 m = 79.2 m^2
What is the product?
(4x)(-3x^8)(-7x^3)
Answers
Answer:
84x¹²
Step-by-step explanation:
To simplify this expression, we work from left to right:
4x(-3x⁸)(-7x³)
Multiplying the coefficients of the first two, we have 4(-3) = -12. Using the laws of exponents, multiplying powers with the same base means we add the exponents; this gives us x(x⁸) = x¹⁺⁸ = x⁹. This gives us
-12x⁹(-7x³)
Multiplying the coefficients, we have -12(-7) = 84. Using the laws of exponents, for the variable we have x⁹(x³) = x⁹⁺³ = x¹²
This gives us
84x¹²
Descent, Inc., produces a variety of climbing and mountaineering equipment. One of its products is a traditional three-strand climbing rope. An important characteristic of any climbing rope is its tensile strength. Descent produces the three-strand rope on two separate production lines: one in Bozeman and the other in Challis. The Bozeman line has recently installed new production equipment. Descent regularly tests the tensile strength of its ropes by randomly selecting them to various tests. The most recent random sample of ropes, taken after the new equipment was installed at the Bozeman plant, revealed the following:
Bozeman; x1= 7,200 lbs
S1=425 n1=25,
Challis;x2=7,087
lbs, S2=415, n2=20
Required:
Conduct the appropriate hypothesis test at the 0.10 level of significance.
Answers
Solution :
Assuming \($\sigma_1^2=\sigma_2^2$\)
We have to test
\($H_0:\mu_1=\mu_2$\)
Against \(H_a: \mu_1 \neq \mu_2\)
Level of significance, \($\alpha = 0.05$\)
\($s_p=\sqrt{\frac{(n_1-1)s_1^2+ (n_2-1)s_2^2}{n_1+n_2-2}}$\)
\($s_p=\sqrt{\frac{(25-1)(425)^2+ (20-1)(415)^2}{25+20-2}}$\)
= 420.6107
Under \(H_0\), the t-statistics is as follows:
\($t=\frac{(\overline{x_1} - \overline{x_2})}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \sim $\) \($\text{t with }(n_1+n_2-2) \ DF$\)
\($t=\frac{(7200-7087)}{(420.6107)\sqrt{\frac{1}{25}+\frac{1}{20}}}$\)
= 0.90
DF = (25 + 20 - 2)
= 43
P-value of the test = 0.375
Since the p value is more than 0.05, we fail to reject our null hypothesis.
There is no difference between then mean tensile strength of the ropes that is produced in the Bozeman and Challis.
Franco, Ivanna, and Esther bought school supplies.
Each person spent between $8 and $13.
Franco bought 6 items.
Ivanna bought 3 items.
Esther bought 4 items.
The prices of the items are given below. Drag items to each box to show what each person could have bought.
Answers
So I think I had this for Imagine Math, so here is what I got :D;
Franco (6 items): 6 notebooks = 1.89+1.89+1.89+1.89+1.89+1.89= 11.34
Ivanna (3 items): 1 colored thingy + 2 sharped thingy = 4.29+2.45+2.45= 9.19
Esther (4 items): 1 colored thingy + notebook + sharpened thingy = 1.89+2.45+0.54+4.29= 9.17
I really hope this helps :DDDDDDDD
hi please help......
Answers
Answer:
9/20
Step-by-step explanation:
45%=45/100
hope this helps
Police body cameras: A survey of New York State residents asked about police officers having to wear video cameras on duty. The question stated, "Do you agree or disagree that police officers should carry video cameras for the purposes of filming their activities while on duty?" Most (88%) respondents agreed with this statement. Do Californians share the same opinion? A California-based civil rights group conducted a similar survey by randomly selecting 500 California residents, and 425 agreed that police officers should carry video cameras for the purposes of filming their activities while on duty. Conduct a hypothesis test to determine if the proportion of California residents who agree is different from New York residents. Use a 5% significance level to make your decision. Use the applet (at the top of this Checkpoint) to determine the P-value. Which of the following is an appropriate conclusion based on the results?
a. The survey provides strong evidence that the proportion of California residents who agree that police officers should carry video cameras for the purposes of filming their activities while on duty is significantly different from the proportion of New York residents.
b. The survey suggests that the proportion of California residents who agree that police officers should carry video cameras for the purposes of filming their activities while on duty is 85%.
c. Of the California residents surveyed, 85% agree that police officers should carry video cameras for the purposes of filming their activities while on duty, but this is not strong enough evidence to conclude that the proportion of California residents who agree is significantly different from the proportion of New York residents.
d. This survey suggests that 88% of California residents surveyed agree that police officers should carry video cameras for the purposes of filming their activities while on duty.
Answers
Answer:
Step-by-step explanation:
Find the slope-intercept form of the line with the given slope and y-intercept.
9
3
Slope = 7: y-intercept = -
8
What is the equation of the line?
Answers
Answer:y=7x+8
Step-by-step explanation:
question is pictured below, need to learn steps to solve this and I dont have my book with me
Answers
However, I can still give you some general steps to solve a math problem in 200 words.
Step 1: Read the problem thoroughly and understand what it's asking you to find. This may involve identifying the given information and what you need to find. Also, identify any constraints on the problem, like time or cost.
Step 2: Plan how to solve the problem. Think about what mathematical operations or formulas you can use to find the answer. Make a list of the steps you need to take to solve the problem. Also, determine the units for your answer.
Step 3: Solve the problem. Use the mathematical operations or formulas you've identified in step 2 to find the answer. Show all of your work, including any calculations, so that you can easily check your work later.
Step 4: Check your answer. Make sure that your answer makes sense and that it satisfies any constraints given in the problem. Also, check that your units are correct. If your answer doesn't make sense, go back and review your work to see if you made an error.
Step 5: Reflect on what you've learned. Think about what you did well and what you could improve upon in future problem-solving situations. This will help you become a more effective problem solver in the future.
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List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8. (b) the set S = (x1x2 + 4x - 5 -0. (c) the set of outcomes when a coin is tossed until a tail or three heads appear. (d) the set S = {x|2x - 4 20 and x < 1)
Answers
a)The set of integers between 1 and 50, divisible by 8 are given as.
S={ 8,16,24,32,40,48}
b ) Hence, the set {-5,1}
c) The list of set of outcomes when a coin is tossed until a tail or three heads appear will be.
S={T,HT,HHT,HHH}
d) S={Europe, Asia, Australia, North America, South America, Africa, Antarctica.}
a) We have to find a list of integers between 1 and 50, divisible by 8.
Keep in mind that the S is the event that represents the sample space. The set of integers between 1 and 50, divisible by 8 are given as.
S={ 8,16,24,32,40,48}
b) We need to list the set S= {\(x| x^2+4x-5=0}\)}
\(x^2+4x-5=0\\\\x^2+5x-x-5=0\\x(x+5)-(x+5)==\\(x+5)(x-1)=0\\\\x=-5 , x=1\)
Hence, the set {-5,1}
c) c) We have to find a list of the set of outcomes when a coin is tossed until a tail or three heads appear.
Let T denote the event that tail appeared.
Let H denote the event that head appeared.
The list of set of outcomes when a coin is tossed until a tail or three heads appear will be.
S={T,HT,HHT,HHH}
d) We need to list the set
S={x ∣ x is a continent}.
There are 7 continents:
S={Europe, Asia, Australia, North America, South America, Africa, Antarctica.}
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What is the percent of 1 - 3√(5/35) ?
Answers
Answer:
1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755
0.0755 * 100 = 7.55%
Step-by-step explanation:
To find the percentage of 1 - 3√(5/35), we need to first evaluate the expression.
1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755
To convert this decimal to a percentage, we simply multiply by 100:
0.0755 * 100 = 7.55%
Help meeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Answers
Step-by-step explanation:
b because it's bbbusjsjs
HELP AGIAN PLEASEEEEEEEEEEEEEE
Answers
Step-by-step explanation:
y=mx+b
6 -8
9 -7
\( - 8 = 6m + b \\ - 7 = 9m + b \\ - 7 + 8 = 9m - 6m \\ 3m= 1 \\ m = \frac{1}{3} \\ \)
\( - 8 = 6 \frac{1}{ 3} + b \\ b = - 8 - 2 \\ b = - 10 \\ y = 0.333x - 10 \\ y = \frac{x}{ 3 } - 10\)
Answer:
i got you
Step-by-step explanation:
to find it use y=mx+b
Which of the following represents a correct rewriting of the following equation, rewritten to solve for y? ax-by>c
Options
A: y < a/b x + c/b
B: y < a/b x - c/b
C: y > a/b x + c/b
D: y > a/b x + c/b
Answers
Answer:
Step-by-step explanation:
Answer is unclear, but keep trying
What is the equation of the line that passes through the point (-1,-5) and has a
slope of -3?
Answers
Answer:
y=-3x-8
Step-by-step explanation:
y-y1=m(x-x1)
y-(-5)=-3(x-(-1))
y+5=-3(x+1)
y=-3x-3-5
y=-3x-8
15 Points and branliest for all three!
Answers
According to SAS Congruence Theorem and the reflexive property of congruence, it can be proved that ΔSPQ ≅ ΔTPQ.
It is given to us that -
PQ bisects ∠SPT
SP ≅ TP
We have to prove that ΔSPQ ≅ ΔTPQ
Now, as PQ bisects ∠SPT,
∠SPQ = ∠TPQ
Also, according to the Reflexive Property of Congruence, PQ is a common side of both triangles - ΔSPQ and ΔTPQ.
Thus, according to SAS Congruence Theorem,
"If two sides and the angle between these two sides are congruent to the corresponding sides and angle of another triangle, then the two triangles are congruent."
Therefore, according to SAS Congruence Theorem, we have proved that ΔSPQ ≅ ΔTPQ.
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Which number line represents the solution set for the inequality 2x - 6 2 6(x - 2) + 8?
Answers
Step-by-step explanation:
2x-6 2 6(x-2)+8 is the answe
Help, please
f(n)=3(n+2)^2 (n-3) (n-2)
As n -----> -oo, f(n)----> ?
As n ----->oo, f(n) ----->?
Answers
When n approaches infinity n -----> -∞ as f(n) -----> -∞ and
as n -----> ∞ then f(n) -----> ∞ for the function f(n) is (n+2)²(n-3)(n-2) then
As n approaches negative infinity, the dominant term of the function f(n) is (n+2)²(n-3)(n-2).
Since (n+2)² grows to positive infinity as n approaches negative infinity, and both (n-3) and (n-2) approach negative infinity, the overall value of f(n) approaches negative infinity.
Therefore, as n -----> -∞ as f(n) -----> -∞
As n approaches positive infinity, the dominant term of the function f(n) is again (n+2)²(n-3)(n-2).
Since (n+2)² grows to positive infinity as n approaches positive infinity, and both (n-3) and (n-2) approach positive infinity, the overall value of f(n) approaches positive infinity.
Therefore, as n -----> ∞, f(n) -----> ∞
Hence, the function f(n) is (n+2)²(n-3)(n-2) then when n approaches infinity n -----> -∞ as f(n) -----> -∞ and as n -----> ∞ then f(n) -----> ∞
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The perimeter of the dining room table shows below is 23 ft what is the missing length. length shown is 4 and 1/4 ft
Answers
Step by step. Assuming the table is rectangular in shape, the one given length of 4-1/4 Ft x 2 sides = 9-1/2 ft.
23 ft - 9-1/2 ft = 13-1/2 ft left for the other two sides.
13-1/2 ft divided by 2 = 6-3/4 Ft is the missing length.
please answer asap.........
Answers
Answer:
1.4 i would wait for someone else to answer it I'm not sure
Step-by-step explanation:
Find an equation of the circle that has center (-3, -6) and passes through (0, 1).
Answers
Answer:
Step-by-step explanation:
General equation for a circle (x - a)² + (y - b)² = r² with (a,b) represents the center, (x,y) represents one of the points lie on the circle, and r represents the radius Determine r² by substituting the points into the general equation (x - a)² + (y - b)² = r² (5 - (-1))² + (-4 - 2)² = r² (5 + 1)² + (-6)² = r² 6² + 36 = r² 36 + 36 = r² 72 = r² Determine the equation of the circle (x - a)² + (y - b)² = r² (x - (-1))² + (y - 2)² = 72 (x + 1)² + (y - 2)² = 72 (This is the equation of the circle)
The triangle above has the following measures.
s = 29 m
r= 63 m
Find the m/Q.
Round to the nearest tenth and include correct units.
Show all your work.
Answers
The value of m∠Q as shown in the right-angled triangle below is 62.59°.
The Formula used is:
m∠Q = cos⁻¹(s/r)........................ (1)
Where:
s = 29 m
r = 63 m
Now, Substitute these values into equation 1
m∠Q = cos⁻¹(29/63)
m∠Q = cos⁻¹(0.46)
m∠Q = 62.59°
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Write down the ratio 240 kg to 6 kg.
Give your answer in its lowest form.
Answers
The ratio of the given 240 kg to 6 kg will be 40:1.
What is Ratio?
Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other. Two categories can be used to categorise ratios. Part to whole ratio is one, and part to part ratio is the other. The part-to-part ratio shows the relationship between two separate entities or groupings. Mathematicians use the term "ratio" to compare two or more numbers. It serves as a comparison tool to show how big or tiny an amount is in relation to another. Two quantities are compared using division in a ratio. In this case, the dividend is referred to as the "antecedent" and the divisor as the "consequent."
the ratio 240 kg to 6 kg.
It will be simple 240/6 :: 40 :1
Hence the ratio of the given data will be 40:1
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Are the following ratios equivalent
16:48 and 112:336
Answers
Answer:
Yes
Step-by-step explanation:
Rick Rich owns a Mercedes dealership. Mercedes has 5 models, 4 standard options packages, and 5 colors. If Rick wants to immediately be able to deliver any car (model, option package, color), how many cars must Rick have on hand?
Answers
Rick would need to have at least 100 cars on hand to immediately be able to deliver any car (model, option package, color).
To immediately deliver any car, Rick Rich's dealership must have all possible combinations of Mercedes models, option packages, and colors in stock.
With five models, four option packages, and five colors, there are 5 x 4 x 5 = 100 possible combinations.
Therefore, Rick would need to have at least 100 cars on hand, each representing a unique combination of model, option package, and color. It's worth noting that having exactly 100 cars on hand would only allow Rick to deliver one of each possible combination, so it may be prudent to have additional inventory on hand to meet demand for more popular combinations.
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In a certain year, a magnitude 7.8 earthquake struck California. In another year, a magnitude 5.8 quake struck Kansas. How many times as intense as the Kansas quake was the California quake?
Answers
Answer:
The California quake was 1.34 times as intense as the Kansas quake.
Step-by-step explanation:
To find how many times as intense as Kansas (K) quake was the California (C) quake first, we need to find the ratio of intensity (I) between those earthquakes:
\( \frac{I_{C}}{I_{K}} = \frac{7.8}{5.8} = 1.34 \)
Now, by multiplying the above number for the intensity of the Kansas quake we should have the intensity of the California quake:
\( I_{K} = 1.34*5.8 = 7.8 \)
Therefore, the California quake was 1.34 times as intense as the Kansas quake.
I hope it helps you!