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Question: If \(\sin 2\theta + \sin 2\varphi = \frac{1}{2}\) and \(\cos 2\theta + \cos 2\varphi = \frac{3}{2}\)...

If sin⁡2θ+sin⁡2φ=12\sin 2\theta + \sin 2\varphi = \frac{1}{2} and cos⁡2θ+cos⁡2φ=32\cos 2\theta + \cos 2\varphi = \frac{3}{2}, then

cos⁡2(θ−φ)\cos^{2}(\theta - \varphi) equal to

A

38\frac{3}{8}

B

58\frac{5}{8}

C

34\frac{3}{4}

D

54\frac{5}{4}

Answer

58\frac{5}{8}

Explanation

Solution

Given, sin⁡2θ+sin⁡2φ=12\sin 2\theta + \sin 2\varphi = \frac{1}{2} .......(i)

and cos⁡2θ+cos⁡2φ=32\cos 2\theta + \cos 2\varphi = \frac{3}{2} .......(ii)

Squaring and adding,

$$$\sin 2\varphi + \cos 2\theta.\cos 2\varphi\rbrack = \frac{1}{4} + \frac{9}{4}$⇒ $\cos 2\theta.\cos 2\varphi + \sin 2\theta.\sin 2\varphi = \frac{1}{4}$ ⇒ $\cos(2\theta - 2\varphi) = \frac{1}{4}$ ⇒ $\cos^{2}(\theta - \varphi) = \frac{5}{8}$.