Question

Which of the following are trigonometric identities? Select all that apply

Answer 1

WSimpe are given the following trigonometric identities:

Part A.

`cos^2\theta=sin^2\theta-1`

This is not a trigonometric identity. The true identity is:

`cos^2\theta+sin^2\theta=1`

Part B

`sin\theta=(1)/(csc\theta)`

This is a trigonometric identity by definition.

Part C.

`sec\theta=(1)/(cot\theta)`

This is not a trigonometric identity by definition

Part D.

`cot\theta=(cos\theta)/(sin\theta)`

This can be proven to be true if we take the following identity:

`cot\theta=(1)/(tan\theta)`

Since

`tan\theta=(sin\theta)/(cos\theta)`

Substituting in the identity for cot we get:

`cot\theta=(1)/((sin\theta)/(cos\theta))`

Simplifying:

`cot\theta=(cos\theta)/(sin\theta)`

Therefore, the identity is true.

Part W.

`1+cot^2\theta=csc^2\theta`

To prove this identity we will use the following identity:

`sin^2\theta+cos^2\theta=1`

Now, we divide both sides by the square of the sine:

`(sin^2\theta)/(sin^2\theta)+(cos^2\theta)/(sin^2\theta)=(1)/(sin^2\theta)`

The first term is 1:

`1+(cos^(2)\theta)/(s\imaginaryI n^(2)\theta)=(1)/(s\imaginaryI n^(2)\theta)`

We can use the identity in part D for the second term:

`1+cot^2\theta=(1)/(sin^2\theta)`

For the last term, we use identity in part B:

`1+cot^2\theta=csc^2\theta`

Therefore, the identity is true.