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

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

This is a trigonometric identity by definition.

Part C.

$sec\theta=\frac{1}{cot\theta}$

This is not a trigonometric identity by definition

Part D.

$cot\theta=\frac{cos\theta}{sin\theta}$

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

$cot\theta=\frac{1}{tan\theta}$

Since

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

Substituting in the identity for cot we get:

$cot\theta=\frac{1}{\frac{sin\theta}{cos\theta}}$

Simplifying:

$cot\theta=\frac{cos\theta}{sin\theta}$

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:

$\frac{sin^2\theta}{sin^2\theta}+\frac{cos^2\theta}{sin^2\theta}=\frac{1}{sin^2\theta}$

The first term is 1:

$1+\frac{cos^{2}\theta}{s\imaginaryI n^{2}\theta}=\frac{1}{s\imaginaryI n^{2}\theta}$

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

$1+cot^2\theta=\frac{1}{sin^2\theta}$

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

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

Therefore, the identity is true.