Verifying a Double-Angle Identity for Tangent

Verify the identity:

\tan (2 θ) = \frac{2}{\cot θ - \tan 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
In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.

$\begin{matrix} \tan (2 θ) & = & \frac{2 \tan θ}{1 - \tan^{2} θ} & Double-angle formula \\ = & \frac{2 \tan θ (\frac{1}{\tan θ})}{(1 - \tan^{2} θ) (\frac{1}{\tan θ})} & Multiply by a term that results in desired numerator . \\ = & \frac{2}{\frac{1}{\tan θ} - \frac{\tan^{2} θ}{\tan θ}} \\ = & \frac{2}{\cot θ - \tan θ} & Use reciprocal identity for \frac{1}{\tan θ} . \end{matrix}$

Your turn

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

Verify the identity: $\cos (2 θ) \cos θ = \cos^{3} θ - \cos θ \sin^{2} θ .$

How did it go?