Question

Use an Addition orSubtraction Formula to find the exact value of the expression, asDemonstrated

$\tan \left(-\frac{\pi }{12}\right)$

Short answer

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

Step-by-step solution

Step 1. Given information.

The given function is$\tan \left(-\frac{\pi }{12}\right)$.

Find out the exact value of the expression.

Then determine the expression to get the answer.

Step 2. Determining the values.

Use the difference formula for the tangent function where:

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

Therefore, $\tan \left(-\frac{\pi }{12}\right)=\tan \left(\frac{\pi }{6}-\frac{\pi }{4}\right)$

$=\frac{\tan \frac{\pi }{6}-\tan \frac{\pi }{4}}{1+\tan \frac{\pi }{6}\tan \frac{\pi }{4}}$

$=\frac{\frac{\sqrt{3}}{3}-1}{1+\frac{\sqrt{3}}{3}\times 1}$

$=-\frac{\sqrt{3}-3}{3+\sqrt{3}}$

$=\frac{\sqrt{3}-3}{\sqrt{3}+3}\times \frac{\sqrt{3}-3}{\sqrt{3}-3}$

$=\frac{{(\sqrt{3}-3)}^2}{{\left(\sqrt{3}\right)}^2-3^2}$

$=\frac{3-6\sqrt{3}+9}{3-9}$

$=-2+\sqrt{3}$