Question

write the following expression as the sine, cosine, or tangent of a double - angle. then find the exact value of the expression.
\\( \frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } } \\)
what is the given expression equal to? select the correct choice below and fill in the answer box within your choice.
(simplify your answer. type an exact answer, using pi as needed. use integers or decimals for any numbers in the expression.)
\\( \frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } } = \sin \\)
\\( \frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } } = \cos \\)
\\( \frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } } = \tan \\)
the exact value of the expression is
(type an exact answer, using radicals as needed. rationalize the denominator.)

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 \(\frac{2\tan\frac{7\pi}{12}}{1-\tan^{2}\frac{7\pi}{12}}\), if we let \(\alpha=\frac{7\pi}{12}\), then by the double - angle formula for tangent, the expression is equal to \(\tan(2\times\frac{7\pi}{12})\).

Step2: Simplify the angle

Calculate \(2\times\frac{7\pi}{12}=\frac{7\pi}{6}\).

Step3: Find the value of \(\tan\frac{7\pi}{6}\)

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for \(\theta = \frac{7\pi}{6}\), \(\sin\frac{7\pi}{6}=-\frac{1}{2}\), \(\cos\frac{7\pi}{6}=-\frac{\sqrt{3}}{2}\).
So \(\tan\frac{7\pi}{6}=\frac{\sin\frac{7\pi}{6}}{\cos\frac{7\pi}{6}}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\).

Answer:

C. \(\frac{2\tan\frac{7\pi}{12}}{1 - \tan^{2}\frac{7\pi}{12}}=\tan\frac{7\pi}{6}\)

The exact value of the expression is \(\frac{\sqrt{3}}{3}\)