Verifying an Identity Involving Tangent

Verify the following identity.

\frac{\sin (α - β)}{\cos α \cos β} = \tan α - \tan β

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
We can begin by rewriting the numerator on the left side of the equation.

$\begin{matrix} \frac{\sin (α - β)}{\cos α \cos β} & = & \frac{\sin α \cos β - \cos α \sin β}{\cos α \cos β} \\ = & \frac{\sin α \cos β}{\cos α \cos β} - \frac{\cos α \sin β}{\cos α \cos β} & Rewrite using a common denominator . \\ = & \frac{\sin α}{\cos α} - \frac{\sin β}{\cos β} & Cancel . \\ = & \tan α - \tan β & Rewrite in terms of tangent . \end{matrix}$
We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.

Your turn

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

Verify the identity: $\tan (π - θ) = - \tan θ .$

How did it go?