Chapter 5, Problem 73RE

To determine

The simplification of expression using the half angle formulas.

Explanation of Solution

Given expression:

  $-\sqrt{\frac{1+\cos 10x}{2}}$

Calculations:

Here the given expression,

  $-\sqrt{\frac{1+\cos 10x}{2}}$ .

  $\begin{array}{l}Now,\\ \text{Acording to half angle formula,}\\ \sin \frac{u}{2}=\pm \sqrt{\frac{1-\cos u}{2}}\\ \cos \frac{u}{2}=\pm \sqrt{\frac{1+\cos u}{2}}\\ \tan \frac{u}{2}=\frac{\sin u}{1+\cos u}\\ \text{now comparing the expression with the formulas,}\\ \cos \frac{10x}{2}=-\sqrt{\frac{1+\cos 10x}{2}}\\ \cos 5x=-\sqrt{\frac{1+\cos 10x}{2}}\\ \text{ The value of all is positive in first quadrant}\text{. The value of cosine is positive in forth quadrant}\\ \text{Therefore,}\\ 0<5x<\frac{3\pi }{2}\end{array}$