Chapter 14, Problem 23MCQ

To determine

The exact value of the expression.

Answer to Problem 23MCQ

The value of the expression is:

  $\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .

Explanation of Solution

Given:

The expression is given as:

  $\cot {75}^{\circ }$

Consider the expression:

  $\cot {75}^{\circ }$

Since,

  $\cot \left({90}^{\circ }-\theta \right)=\tan \theta$

Therefore,

  $\begin{array}{l}\cot {75}^{\circ }=\cot \left({90}^{\circ }-{15}^{\circ }\right)\\ =\tan {15}^{\circ }\end{array}$

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