Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Verify the identity $\csc^{2} θ - 2 = \frac{\cos (2 θ)}{\sin^{2} θ} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

$\begin{matrix} \frac{\cos (2 θ)}{\sin^{2} θ} & = & \frac{1 - 2 \sin^{2} θ}{\sin^{2} θ} \\ = & \frac{1}{\sin^{2} θ} - \frac{2 \sin^{2} θ}{\sin^{2} θ} \\ = & \csc^{2} θ - 2 \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Verify the identity $\tan θ \cot θ - \cos^{2} θ = \sin^{2} θ .$

How did it go?