2 2 + ^2( - )- ^2( + ) is equal to: 1. 2( + ) 2. 2( + ) 3. 2( - )… - Vidyadip

Question

$\cos2\theta\cos2\phi+\sin^2(\theta-\phi)-\sin^2(\theta+\phi)$ is equal to:

$\sin2(\theta+\phi)$
$\cos2(\theta+\phi)$
$\sin2(\theta-\phi)$
$\cos2(\theta-\phi)$

[Hint: Use $\sin^2\text{A}-\sin^2\text{B}=\sin(\text{A+B})\sin(\text{A}-\text{B})$]

✓

Answer

$\cos2(\theta+\phi)$

Solution:

$\cos2\theta\cos2\phi+\sin^2(\theta-\phi)-\sin^2(\theta+\phi)$

$=\cos2\theta\cos2\phi+\sin(\theta-\phi+\theta+\phi)\sin(\theta-\phi-\theta-\phi)$

$=\cos2\theta\cos2\phi-\sin2\theta\sin2\phi=\cos(2\theta+2\phi)=\cos2(\theta+\phi)$

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