Chapter 5, Problem 116RE

To determine

Tocalculate:The simplified expression by using the half-angle formulas.

Answer to Problem 116RE

The simplified expression is $\tan \left(6x\right)$ .

Explanation of Solution

Given information:

The expression $=\frac{1-\cos 12x}{\sin 12x}$

Formula used:

The half-angle formulas for trigonometric identities are shown below:

  $\begin{array}{l}\sin \left(\frac{u}{2}\right)=\pm \sqrt{\frac{1-\cos u}{2}}\\ \cos \left(\frac{u}{2}\right)=\pm \sqrt{\frac{1+\cos u}{2}}\\ \tan \left(\frac{u}{2}\right)=\frac{1-\cos u}{\sin u}=\frac{\sin u}{1+\cos u}\end{array}$

Calculation:

Know that, $\tan \left(\frac{u}{2}\right)=\frac{1-\cos u}{\sin u}$

Here, it is seen that $u=12x$ ;

So, $\frac{u}{2}=\frac{12x}{2}=6x$

Substitute $12x$ for $u$ and $6x$ for $\frac{u}{2}$ in the expression $\tan \left(\frac{u}{2}\right)=\frac{1-\cos u}{\sin u}$ .

  $\tan \left(6x\right)=\frac{1-\cos 12x}{\sin 12x}$

Hence, the required simplified expression is $\tan \left(6x\right)$ .