Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.
-$$\frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } }$$
A) $\frac { \sqrt { 3 } } { 3 }$
B) $- \frac { \sqrt { 3 } } { 3 }$
C) $\sqrt { 3 }$
D) $- \sqrt { 3 }$

Question

Quizplus · Accepted Answer

The expression is in the form of the tangent of a double angle, so we can write it as:$$\frac { 2 	an \frac { 7 \pi } { 12 } } { 1 - 	an ^ { 2 } \frac { 7 \pi } { 12 } } = 	an 2 \cdot \frac { 7 \pi } { 12 }$$Now, we can use the double-angle formula for tangent to find the exact value of the expression:$$	an 2 \cdot \frac { 7 \pi } { 12 } = \frac { 2 	an \frac { 7 \pi } { 12 } } { 1 - 	an ^ { 2 } \frac { 7 \pi } { 12 } } = \frac { 2 \cdot \left( - \frac { 1 } { \sqrt { 3 } } ight) } { 1 - \left( - \frac { 1 } { \sqrt { 3 } } ight) ^ { 2 } } = \frac { - \frac { 2 } { \sqrt { 3 } } } { 1 - \frac { 1 } { 3 } } = \frac { - \frac { 2 } { \sqrt { 3 } } } { \frac { 2 } { 3 } } = - \frac { \sqrt { 3 } } { 3 }$$Therefore, the correct answer is A.

Write the Expression as the Sine, Cosine, or Tangent of a Double