Chapter 7, Problem 45RE

To determine

To establish:the given identity.

Answer to Problem 45RE

The identity is established.

Explanation of Solution

Given:

The given identity is $5{\cos }^2\theta +3si{n}^2\theta =3+2{\cos }^2\theta$ .

Calculation:

As the given identity is $5{\cos }^2\theta +3si{n}^2\theta =3+2{\cos }^2\theta$ .

Now, let us check LHS:

  $\begin{array}{l}LHS=5{\cos }^2\theta +3si{n}^2\theta \\ =2{\cos }^2\theta +3{\cos }^2\theta +3si{n}^2\theta \\ =2{\cos }^2\theta +3\left({\cos }^2\theta +si{n}^2\theta \right)\\ \left\{since,\left({\cos }^2\theta +si{n}^2\theta \right)=1\right\}\\ \Rightarrow LHS=2{\cos }^2\theta +3\left(1\right)\\ =3+2{\cos }^2\theta \\ \text{Also},RHS=3+2{\cos }^2\theta \end{array}$

Thus, $5{\cos }^2\theta +3si{n}^2\theta =3+2{\cos }^2\theta$

Hence, the identity is established.