Chapter 7.1, Problem 1CFU

To determine

To find: a counterexample to show that the equation is not an identity.

Answer to Problem 1CFU

  $1-\sin \frac{\pi }{6}\ne \cos \frac{\pi }{6}$

Explanation of Solution

Given information:

  $1-\sin x=\cos x$

Calculation:

Let $x=\frac{\pi }{6}$ . Put the value of x in the given equation and simplify as shown below.

  $\begin{array}{c}1-\sin \frac{\pi }{6}=\cos \frac{\pi }{6}\\ 1-\frac{1}{2}=\frac{\sqrt{3}}{2}\\ \frac{1}{2}\ne \frac{\sqrt{3}}{2}\end{array}$

Thus the given equation is not a trigonometric identity.