Question

If $1+\cos x\cos y+\sin x\sin y=0$, then-

• A. $\cos x-\cos y=0$
• B. $\cos x+\cos y=0$
• C. $\sin x+\sin y=0$
• D. $\cos x+\sin y=1$

Answer 1

Answer: (B), (C)

$\cos x+\cos y=0$
$\sin x+\sin y=0$

Given $1+\cos x\cos y+\sin x\sin y=0$

As we know that $\cos (a-b)=\cos a\cos b+\sin a\sin b$

$⟹\cos (x-y)=-1$

$⟹x-y=(2k+1)\pi$ where $k$ is an integer

$\cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}=2\cos \frac{x+y}{2}\cos \frac{(2k+1)\pi }{2}=0$ $(\because \cos \frac{n\pi }{2}=0,$ for any integer $n)$

$\sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2}=2\sin \frac{x+y}{2}\cos \frac{(2k+1)\pi }{2}=0$