Chapter 7.4, Problem 30E

To determine

To verify: that the given equation is an identity.

Explanation of Solution

Given information:

  ${\left(\sin \theta +\cos \theta \right)}^2-1=\sin 2\theta$

Calculation:

Simplify the left side of the equation into right side as follows:

  $\begin{array}{c}{\left(\sin \theta +\cos \theta \right)}^2-1=\sin 2\theta \\ {\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta -1=\sin 2\theta \\ 1+2\sin \theta \cos \theta -1=\sin 2\theta \left[\because {\sin }^2\theta +{\cos }^2\theta =1\right]\\ 2\sin \theta \cos \theta =\sin 2\theta \\ \sin 2\theta =\sin 2\theta \\ LHS=RHS\end{array}$

Since the left side of the equation is transformed into right hand side. So, the equation is an identity.