Question

The value of $2{\sin }^2\theta +4\cos (\theta +\alpha )\sin \alpha \sin \theta +\cos 2(\alpha +\theta )$ is

• A. $\cos \theta +\cos \alpha$
• B. Independent of $\theta$
• C. Independent of $\alpha$
• D. Independent of both $\theta$ and $\alpha$

Answer 1

Answer: (B)

Independent of $\theta$

$2{\sin }^2\theta +4\cos (\theta +\alpha )\sin \alpha \sin \theta +\cos 2(\theta +\alpha )$

$=2{\sin }^2\theta +4\cos (\theta +\alpha )\sin \alpha \sin \theta +2{\cos }^2(\theta +\alpha )-1$

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

$=2{\sin }^2\theta +2\cos (\theta +\alpha )(2\sin \alpha \sin \theta +\cos \theta \cos \alpha -\sin \theta \sin \alpha )-1$

$=2{\sin }^2\theta +2\cos (\theta +\alpha )(\cos \theta \cos \alpha +\sin \theta \sin \alpha )-1$

$=2{\sin }^2\theta +2\cos (\theta +\alpha )\cos (\theta -\alpha )-1$

$=2{\sin }^2\theta +\cos 2\theta +\cos 2\alpha -1$

$=2si{n}^2\theta +1-2{\sin }^2\theta +\cos 2\alpha -1=\cos 2\alpha$