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  4. The value of 2( sin)2θ +4 cos(θ + α ) sinα sinθ + cos2 (α + θ ) is

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Q. The value of $2\left(\sin\right)^{2}\theta +4\cos\left(\theta + \alpha \right)\sin\alpha \sin\theta +\cos2 \left(\alpha + \theta \right)$ is

NTA AbhyasNTA Abhyas 2022

Solution:

$2 \sin ^2 \theta+2 \cos (\theta+\alpha)(\cos (\theta-\alpha)-\cos (\theta+\alpha))+\left(2 \cos ^2(\theta+\alpha)-1\right)$
$=2 \sin ^2 \theta+2\left(\cos ^2 \theta-\sin ^2 \alpha\right)-1$
$=2(\sin )^2 \theta+2\left((\cos )^2 \theta-(\sin )^2 \alpha\right)-1$
$=2-2 \sin ^2 \alpha-1=\cos ^2 \alpha-\sin ^2 \alpha$