Question

write the following expression as the sine, cosine, or tangent of a double - angle. then find the ex expression.
\\( \frac { 2 \tan \frac { \pi } { 6 } } { 1 - \tan ^ { 2 } \frac { \pi } { 6 } } \\)
the expression.)
\\( \frac { 2 \tan \frac { \pi } { 6 } } { 1 - \tan ^ { 2 } \frac { \pi } { 6 } } = \sin \\)
\\( \frac { 2 \tan \frac { \pi } { 6 } } { 1 - \tan ^ { 2 } \frac { \pi } { 6 } } = \tan \\)
\\( \frac { 2 \tan \frac { \pi } { 6 } } { 1 - \tan ^ { 2 } \frac { \pi } { 6 } } = \cos \\)

Explanation:

Step1: Recall the double - angle formula for tangent

The double - angle formula for tangent is \(\tan(2\alpha)=\frac{2\tan\alpha}{1 - \tan^{2}\alpha}\).
In the given expression \(\frac{2\tan\frac{\pi}{6}}{1-\tan^{2}\frac{\pi}{6}}\), if we let \(\alpha=\frac{\pi}{6}\), then by the double - angle formula for tangent \(\frac{2\tan\frac{\pi}{6}}{1-\tan^{2}\frac{\pi}{6}}=\tan(2\times\frac{\pi}{6})\).

Step2: Simplify the angle

Calculate \(2\times\frac{\pi}{6}=\frac{\pi}{3}\).

Answer:

B. \(\frac{2\tan\frac{\pi}{6}}{1 - \tan^{2}\frac{\pi}{6}}=\tan\frac{\pi}{3}\)