Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as Demonstrated.

$\tan \frac{7\pi }{12}$

Short answer

Therefore, the exact value of the expression is $\mathbf{-}\mathbf{2}\mathbf{-}\sqrt{\mathbf{3}}$.

Step-by-step solution

Step 1. Given information.

The given function is, $\tan \frac{7\pi }{12}$

Step 2. Determining the value.

Use the sum formula for the tangent function where

$\tan (a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$

Therefore, $\tan \frac{7\pi }{12}=\tan \left(\frac{\pi }{4}+\frac{\pi }{3}\right)$

$$\begin{gathered} =\frac{\tan \frac{\pi }{4}+\tan \frac{\pi }{3}}{1-\tan \frac{\pi }{4}\tan \frac{\pi }{3}} \\ =\frac{1+\sqrt{3}}{1-1\times \sqrt{3}} \\ =\frac{1+\sqrt{3}}{1-\sqrt{3}} \\ =\frac{1+\sqrt{3}}{1-\sqrt{3}}\times \frac{1+\sqrt{3}}{1+\sqrt{3}} \\ =\frac{(1+{\sqrt{3)}}^2}{1^2-3^2} \\ =\frac{1+2\sqrt{3}+3}{1-3} \\ =\frac{4+2\sqrt{3}}{-2}=-2-\sqrt{3} \end{gathered}$$