Q: Which of the following is the correct definition of $$\csc \theta$$?

$$\csc \theta = \frac{1}{\sin \theta}$$

Q: If $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$, what is $$\cot \theta + 1 in $$terms of $$\csc \theta$$?

$$\cot \theta + 1 = \csc \theta (\cos \theta + \sin \theta)$$

Q: Which identity is most useful for expressing all trigonometric functions in terms of sine and cosine?

$$\tan x = \frac{\sin x}{\cos x}$$

Question 1

Which of the following is the correct fundamental identity relating tangent and secant functions?

Accepted Answer

$$	an^2$ x + 1 = \sec^2$ x$$

Question 2

If $$\sin^2$ x + 2\sin x + 1 is $$factored, which expression is equivalent?

Accepted Answer

$$(\sin x + 1)^2$

Question 3

Which substitution is most helpful when verifying identities involving $$1 + \sin x in $$the denominator?

Accepted Answer

Multiply numerator and denominator by $$1 - \sin x$$

Question 4

What is the correct expression for $$\sec x in $$terms of cosine?

Accepted Answer

$$\sec x = \frac{1}{\cos x}$$

Question 5

Which identity is used to simplify $$1 - \sin^2$ x$$?

Accepted Answer

$$\cos^2$ x = 1 - \sin^2$ x$$

Question 6

In electronics, the total energy $$E(t) in a $$radio circuit is given by $$E(t) = k \cos^2$(2\pi Ft) + k \sin^2$(2\pi Ft). $$Using trigonometric identities, what is $$E(t)$$?

Accepted Answer

$$E(t) = k$$

Question 7

Which of the following is NOT a valid strategy for verifying trigonometric identities?

Accepted Answer

Multiplying both sides of the equation by the same term

Question 8

Given $$	an^2$ x + 1 = \frac{1}{\cos^2$ x}$$, which identity can be used to verify this equation?

Accepted Answer

$$	an^2$ x + 1 = \sec^2$ x$$

Question 9

Which of the following is the correct definition of $$\csc 	heta$$?

Accepted Answer

$$\csc 	heta = \frac{1}{\sin 	heta}$$

Question 10

If $$\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$$, what is $$\cot 	heta + 1 in $$terms of $$\csc 	heta$$?

Accepted Answer

$$\cot 	heta + 1 = \csc 	heta (\cos 	heta + \sin 	heta)$$

Question 11

Which of the following is the correct definition of $$\cot x$$?

Accepted Answer

$$\cot x = \frac{1}{	an x}$$

Question 12

Which identity is used to verify $$\frac{\cos x}{1 - \sin x} = \frac{1 + \sin x}{\cos x}$$?

Accepted Answer

$$\cos^2$ x = 1 - \sin^2$ x$$

Question 13

Which of the following is a valid technique for verifying identities by working with both sides?

Accepted Answer

Transform each side independently into a common third expression

Question 14

If $$	an t = \frac{\sin t}{\cos t}$$ and $$\cot t = \frac{\cos t}{\sin t}$$, what is $$\frac{	an t - \cot t}{\sin t \cos t}$$ simplified?

Accepted Answer

$$\sec^2$ t - \csc^2$ t$$

Question 15

Which identity is most useful for expressing all trigonometric functions in terms of sine and cosine?

Accepted Answer

$$	an x = \frac{\sin x}{\cos x}$$

Question 16

Which of the following is the correct expression for $$	an^2$ x (1 + \cot^2$ x)$$?

Accepted Answer

$$\frac{1}{1 - \sin^2$ x}$$

Question 17

If $$\cos^2$ x = 1 - \sin^2$ x$$, what is $$\frac{1}{1 - \sin^2$ x}$$ equivalent to?

Accepted Answer

$$\sec^2$ x$$

Question 18

A radio circuit stores energy in an inductor and a capacitor, given by $$E(t) = k \cos^2$(2\pi Ft) + k \sin^2$(2\pi Ft). $$Which trigonometric identity proves that $$E(t) is $$constant?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Study Guide: Verifying Trigonometric Identities

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