Chapter 7.1, Problem 21E

To determine

To proof: that the equation is not a trigonometric identity by giving counterexample.

Answer to Problem 21E

  $\sin \frac{\pi }{4}+\cos \frac{\pi }{4}\ne 1$

Explanation of Solution

Given information:

  $\sin x+\cos x=1$

Calculation:

Let $x=\frac{\pi }{4}$ and solve the given equation for this value of x .

  $\begin{array}{c}\sin x+\cos x=1\\ \sin \frac{\pi }{4}+\cos \frac{\pi }{4}=1\left[\text{Replace }x\text{ with }\frac{\pi }{4}\right]\\ \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=1\\ \frac{2\sqrt{2}}{2}=1\\ \sqrt{2}\ne 1\end{array}$

Since evaluating each side of the equation for same value of $x$ produces an inequality, so the equation is not an identity.