Question

The general value of $x$ for which $\cos 2x$, $\frac{1}{2}$ and $\sin 2x$ are in AP are given by

• A. $n\pi ,n\pi +\frac{\pi }{2}$
• B. $n\pi ,n\pi +\frac{\pi }{4}$
• C. $n\pi +\frac{\pi }{4}$
• D. $n\pi$

Answer 1

Answer: (B)

$n\pi ,n\pi +\frac{\pi }{4}$

Given:
$\cos 2x+\sin 2x=1$
$\Rightarrow \sin 2x=1-\cos 2x$
$\Rightarrow {\sin }^{2}2x=1+{\cos }^{2}2x-2\cos 2x$
$\Rightarrow 1-{\cos }^{2}2x=1+{\cos }^{2}2x-2\cos 2x$
$\Rightarrow 0={\cos }^{2}2x-\cos 2x$
$\Rightarrow \cos 2x(\cos 2x-1)=0$
$\Rightarrow \cos 2x=0$ or $\cos 2x=1$
$\Rightarrow 2x=2n\pi +\frac{\pi }{2}or2x=2n\pi +0$
$\Rightarrow x=n\pi +\frac{\pi }{4}$ or $x=n\pi$