( + )^2+( + )^2= - Vidyadip

MCQ

$(\cos \alpha+\cos \beta)^2+(\sin \alpha+\sin \beta)^2=$

✓
$4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)$
B
$4 \sin ^2\left(\frac{\alpha-\beta}{2}\right)$
C
$4 \cos ^2\left(\frac{\alpha+\beta}{2}\right)$
D
$4 \sin ^2\left(\frac{\alpha+\beta}{2}\right)$

✓

Answer

Correct option: A.

$4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)$

(A)
$(\cos \alpha+\cos \beta)^2+(\sin \alpha+\sin \beta)^2$
$\begin{array}{l}=\cos ^2 \alpha+\cos ^2 \beta+2 \cos \alpha \cos \beta+\sin ^2 \alpha \\ +\sin ^2 \beta+2 \sin \alpha \sin \beta\end{array}$
$=2\{1+\cos (\alpha-\beta)\}=4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)$

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