# If Sin 2 θ + Sin 2 ϕ = 1 2 and Cos 2 θ + Cos 2 ϕ = 3 2 , Then Cos2 (θ − ϕ) = - Mathematics

MCQ

If sin 2 θ + sin 2 ϕ = $\frac{1}{2}$ and cos 2 θ + cos 2 ϕ = $\frac{3}{2}$, then cos2 (θ − ϕ) =

#### Options

• $\frac{3}{8}$

• $\frac{5}{8}$

• $\frac{3}{4}$

• $\frac{5}{4}$

#### Solution

$\frac{5}{8}$
Given:
sin 2θ + sin 2ϕ = $\frac{1}{2}$                  .....(i)
and
cos 2θ + cos 2ϕ = $\frac{3}{2}$          .....(ii)
Squaring and adding (i) and (ii), we get:
(sin 2θ + sin 2ϕ)2 + (cos 2θ + cos 2ϕ)2 = $\frac{1}{4} + \frac{9}{4}$
$\Rightarrow \left[ 2\sin\left( \frac{2\theta + 2\phi}{2} \right)\cos\left( \frac{2\theta - 2\phi}{2} \right) \right]^2 + \left[ 2\cos\left( \frac{2\theta + 2\phi}{2} \right)\cos\left( \frac{2\theta - 2\phi}{2} \right) \right]^2 = \frac{5}{2}$
$\Rightarrow 4 \sin^2 \left( \theta + \phi \right) \cos^2 \left( \theta - \phi \right) + 4 \cos^2 \left( \theta + \phi \right) \cos^2 \left( \theta - \phi \right) = \frac{5}{2}$
$\Rightarrow 4 \cos^2 \left( \theta - \phi \right)\left[ \sin^2 \left( \theta + \phi \right) + \cos^2 \left( \theta + \phi \right) \right] = \frac{5}{2}$
$\Rightarrow 4 \cos^2 \left( \theta - \phi \right) = \frac{5}{2}$
$\Rightarrow \cos^2 \left( \theta - \phi \right) = \frac{5}{8}$
Concept: Transformation Formulae
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 8 Transformation formulae
Q 3 | Page 21