Chapter 5, Problem 114RE

To determine

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

Answer to Problem 114RE

The simplified expression is $\cos \left(3x\right)$ .

Explanation of Solution

Given information:

The expression $=\sqrt{\frac{1-\cos 6x}{2}}$

Formula used:

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

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

Calculation:

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

Take modulus of the above formula to get the positive expression.

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

Here, it is seen that $u=6x$ ;

So, $\frac{u}{2}=\frac{6x}{2}=3x$

Substitute $6x$ for $u$ and $3x$ for $\frac{u}{2}$ in the expression $\sqrt{\frac{1-\cos u}{2}}=\sin \left(\frac{u}{2}\right)$ .

  $\sqrt{\frac{1-\cos 6x}{2}}=\cos \left(3x\right)$

Hence, the required simplified expression is $\cos \left(3x\right)$ .