Question

If $\sin \beta$ is the G.M between $\sin \alpha$ and $\cos \alpha$ then $\cos 2\beta$ is equal to:

• A. $2{\sin }^2(\frac{\pi }{4}-\alpha )$
• B. $2{\cos }^2(\frac{\pi }{4}-\alpha )$
• C. $2{\cos }^2(\frac{\pi }{4}+\alpha )$
• D. $2{\sin }^2(\frac{\pi }{4}+\alpha )$

Answer 1

Answer: (A), (C)

The correct options are
A $2{\sin }^2(\frac{\pi }{4}-\alpha )$
C $2{\cos }^2(\frac{\pi }{4}+\alpha )$

${\sin }^2\beta =\sin \alpha \cos \alpha$
$\frac{1-\cos 2\beta }{2}=\sin \alpha \cos \alpha$
$\Rightarrow \cos 2\beta =1-2\sin \alpha \cos \alpha$
$\Rightarrow \cos 2\beta =1-\sin 2\alpha$
$\Rightarrow \cos 2\beta =1-\cos (\frac{\pi }{2}-2\alpha )$ or $\cos 2\beta =1+\cos (\frac{\pi }{2}+2\alpha )$
$=2{\sin }^2(\frac{\pi }{4}-\alpha )$ or $2{\cos }^2(\frac{\pi }{4}+\alpha )$