Chapter 7.1, Problem 6CFU

To determine

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

Answer to Problem 6CFU

  $\sin 45°+\cos 45°\ne \tan 45°$

Explanation of Solution

Given information:

  $\sin \theta +\cos \theta =\tan \theta$

Calculation:

Let $\theta =45°$ and solve the given equation for this value of $\theta$ .

  $\begin{array}{c}\sin \theta +\cos \theta =\tan \theta \\ \sin 45°+\cos 45°=\tan 45°\left[\text{Replace }\theta \text{ with 45}°\right]\\ \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=1\\ \frac{2\sqrt{2}}{2}=1\\ \sqrt{2}\ne 1\end{array}$

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