Chapter 13.3, Problem 22MCQ

To determine

The exact value of the expression.

Answer to Problem 22MCQ

The value of the expression is:

  $\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .

Explanation of Solution

Given:

The expression is given as:

  $\tan {15}^{\circ }$

Consider the expression.

  $\tan {15}^{\circ }$

From difference identify:

  $\tan \left(A-B\right)=\frac{\tan A-\tan B}{1+\tan A\tan B}$

Therefore,

  $\begin{array}{l}\tan \left({45}^{\circ }-{30}^{\circ }\right)=\frac{\tan {45}^{\circ }-\tan {30}^{\circ }}{1+\tan {45}^{\circ }\tan {30}^{\circ }}\\ =\frac{1-\left(\frac{1}{\sqrt{3}}\right)}{1+1.\left(\frac{1}{\sqrt{3}}\right)}\\ =\frac{\sqrt{3}-1}{\sqrt{3}+1}\end{array}$

Here, solution of the expression is equal to $\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .