Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\tan 75 ^ { \circ }$
A) $2 - \sqrt { 3 }$
B) $- 2 + \sqrt { 3 }$
C) $- 2 - \sqrt { 3 }$
D) $2 + \sqrt { 3 }$

Question

Quizplus · Accepted Answer

The half-angle formula for tangent is $\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$. For $\tan 75^\circ$, we can use $\theta = 150^\circ$ in the formula. $\cos 150^\circ = -\sqrt{3}/2$, so $\tan 75^\circ = \sqrt{\frac{1 - (-\sqrt{3}/2)}{1 + (-\sqrt{3}/2)}} = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$. Rationalizing the denominator gives $2 + \sqrt{3}$.

Use the Half-Angle Formulas to Find the Exact Value of the Trigonometric