Question

If $\sin 2\theta +\sin 2\varphi =\frac{1}{2}$and $\cos 2\theta +\cos 2\varphi =\frac{3}{2}$ then ${\cos }^2(\theta -\varphi )$:

• A. $\frac{3}{8}$
• B. $\frac{5}{8}$
• C. $\frac{3}{4}$
• D. $\frac{5}{4}$

Answer 1

The correct option is B

$\frac{5}{8}$

Explanation for the correct option.

Step 1: Simplify the equation

Given that, $\sin 2\theta +\sin 2\varphi =\frac{1}{2}$ and $\cos 2\theta +\cos 2\varphi =\frac{3}{2}$

Now using $\sin a+\sin b=2\sin \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)$ and $\cos a+\cos b=2\cos \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)$.

$$\begin{gathered} \Rightarrow 2\sin (\theta +\varphi )\cos (\theta -\varphi )=\frac{1}{2}\dots \text{(I)} \\ \Rightarrow 2\cos (\theta +\varphi )\cos (\theta -\varphi )=\frac{3}{2}\dots \text{(II)} \end{gathered}$$

Step 2: Find the value of $\theta +\phi$

Divide equation (I) by equation (II).
$$\begin{gathered} \tan (\theta +\varphi )=\frac{1}{3} \\ \Rightarrow {\cos }^2(\theta +\varphi )=\frac{9}{10} \\ \Rightarrow \cos (\theta +\varphi )=\frac{3}{\sqrt{10}} \end{gathered}$$
Substitute value of $\cos \left(\theta +\phi \right)$ in equation (II) we get:

$\begin{array}{rcl}\therefore 2·\frac{3}{\sqrt{10}}·\cos (\theta -\varphi )& =& \frac{3}{2}\\ c\mathrm{os}(\theta -\varphi )& =& \frac{\sqrt{10}}{4}\\ co{s}^2(\theta -\varphi )& =& \frac{10}{16}\\ & =& \frac{5}{8}\end{array}$
Hence, option B is correct.