Find the exact value of the expression.
-$\tan 75 ^ { \circ }$
A) $\sqrt { 3 } + 2$
B) $\sqrt { 3 } - 2$
C) $- \sqrt { 3 } - 2$
D) $- \sqrt { 3 } + 2$

Question

Quizplus · Accepted Answer

The exact value of $\tan 75^\circ$ can be found using the angle sum identity for tangent, where $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$. Using $\tan 45^\circ = 1$ and $\tan 30^\circ = \frac{\sqrt{3}}{3}$, we get $\tan 75^\circ = \tan (45^\circ + 30^\circ) = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \cdot \frac{\sqrt{3}}{3}} = \frac{\sqrt{3} + 3}{3 - \sqrt{3}}$. Multiplying numerator and denominator by the conjugate of the denominator, we get $\frac{(\sqrt{3} + 3)(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} = \frac{3\sqrt{3} + 9 + 3\sqrt{3} + 3}{9 - 3} = \frac{6\sqrt{3} + 12}{6} = \sqrt{3} + 2$.

Find the Exact Value of the Expression tan⁡75∘\tan 75 ^ { \circ }tan75∘

Find the Exact Value of the Expression $\tan 75 ^ { \circ }$