Question

use an identity to simplify the following expression.
\frac{\tan 13^{\circ}}{1 - \tan ^{2} 13^{\circ}}
choose the correct expression equal to \frac{\tan 13^{\circ}}{1 - \tan ^{2} 13^{\circ}} below.
\bigcirc a. \frac{1}{2} \tan 13^{\circ}
\bigcirc b. \frac{1}{2} \tan 26^{\circ}
\bigcirc c. \sin 26^{\circ}
\bigcirc d. \frac{1}{2} \cos 13^{\circ}

Explanation:

Step1: Recall the double - angle formula for tangent

The double - angle formula for tangent is \(\tan2\alpha=\frac{2\tan\alpha}{1 - \tan^{2}\alpha}\).
We can rewrite it as \(\frac{\tan\alpha}{1-\tan^{2}\alpha}=\frac{1}{2}\tan2\alpha\).

Step2: Substitute \(\alpha = 13^{\circ}\) into the formula

When \(\alpha = 13^{\circ}\), we have \(\frac{\tan13^{\circ}}{1-\tan^{2}13^{\circ}}\).
Using the formula \(\frac{\tan\alpha}{1-\tan^{2}\alpha}=\frac{1}{2}\tan2\alpha\), substituting \(\alpha = 13^{\circ}\) gives \(\frac{1}{2}\tan(2\times13^{\circ})\).
Since \(2\times13^{\circ}=26^{\circ}\), the expression becomes \(\frac{1}{2}\tan26^{\circ}\).

Answer:

B. \(\frac{1}{2}\tan26^{\circ}\)