Question

Which of the following identities are true? Select all that apply. 3 answers

Answer 1

The correct identities are:

A. csc = 1/sin θ

C. 1+cot² = csc²θ

E. cos(-x) = cos x

True identity of trigonometric functions

Here's the analysis of the given trigonometric identities:

A. csc = 1/sin θ: True.

csc(θ) = 1/sin(θ) is the reciprocal identity of sine.

B. cot = sin θ/cos θ:

False.

cot(θ) = cos(θ)/sin(θ) is the ratio identity of cotangent.

C. 1+cot² = csc²theta: True.

This is the Pythagorean identity for trigonometric ratios.

D. sin(-x) = sin x: False.

sin(-x) = -sin(x) is the odd-angle identity for sine.

E. cos(-x) = cos x: True.

cos(-x) = cos(x) is the even-angle identity for cosine.

Therefore, the correct identities are:

A. csc = 1/sin θ

C. 1+cot² = csc²θ

E. cos(-x) = cos x

Answer 2

Step 1

Trigonometric inverse theorem

`csc\theta\text{ = }(1)/(sin\theta)\text{ TRUE}`

Step 2

`\begin{gathered} Trigonometric\text{ ratio} \\ cot\theta\text{ = }(cos\theta)/(sin\theta) \\ Option\text{ B is FALSE} \end{gathered}`

Step 3

`\begin{gathered} 1\text{ + cot}^2\theta \\ =\text{ 1 + }(cos^2\theta)/(sin^2\theta) \\ =\text{ }(sin^2\theta+cos^2\theta)/(sin^2\theta) \\ \text{= }(1)/(sin^2\theta) \\ \text{= csc}^2\theta \\ 1\text{ + cot}^2\theta\text{ = csc}^2\theta\text{ TRUE} \end{gathered}`

Step 4

`\begin{gathered} sin(-\pi) \\ =\text{ sin\lparen0-}\pi) \\ =\text{ sin0cos}\pi\text{ - sin}\pi cos0 \\ =\text{ -sin}\pi \\ sin(-\pi)\text{ = -sin}\pi \\ Option\text{ D is FALSE} \end{gathered}`

Step 5

`\begin{gathered} cos(-\pi) \\ =\text{ cos\lparen0-}\pi) \\ =\text{ cos0cos}\pi\text{ + sin0sin}\pi \\ =\text{ cos}\pi \\ cos(-\pi)\text{ = cos}\pi\text{ TRUE} \end{gathered}`