Question

use an identity to simplify the following expression.
\frac{\tan 17^{\circ}}{1 - \tan ^{2} 17^{\circ}}

choose the correct expression equal to \frac{\tan 17^{\circ}}{1 - \tan ^{2} 17^{\circ}} below.

\\( \bigcirc \\) a. \\( \frac{1}{2} \tan 17^{\circ} \\)
\\( \bigcirc \\) b. \\( \frac{1}{2} \cos 17^{\circ} \\)
\\( \bigcirc \\) c. \\( \frac{1}{2} \tan 34^{\circ} \\)
\\( \bigcirc \\) d. \\( \sin 34^{\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 \(\alpha = 17^{\circ}\) into the formula

When \(\alpha = 17^{\circ}\), we have \(\frac{\tan17^{\circ}}{1-\tan^{2}17^{\circ}}\).
Using the formula \(\frac{\tan\alpha}{1-\tan^{2}\alpha}=\frac{1}{2}\tan2\alpha\), substituting \(\alpha = 17^{\circ}\) gives \(\frac{1}{2}\tan(2\times17^{\circ})\).
Since \(2\times17^{\circ}=34^{\circ}\), the expression \(\frac{\tan17^{\circ}}{1 - \tan^{2}17^{\circ}}=\frac{1}{2}\tan34^{\circ}\).

Answer:

C. \(\frac{1}{2}\tan34^{\circ}\)