Question

The value of $\cos y\cos \left(\frac{\pi }{2}-x\right)-\cos \left(\frac{\pi }{2}-y\right)\cos x$ $+\sin y\cos \left(\frac{\pi }{2}-x\right)+\cos x\sin \left(\frac{\pi }{2}-y\right)$ is zero, if

• A. $x=0$
• B. $y=0$
• C. $x=y$
• D. None of these

Answer 1

Answer: (D)

We have, $\cos y\cos \left(\frac{\pi }{2}-x\right)-\cos \left(\frac{\pi }{2}-y\right)\cos x$
$+\sin y\cos \left(\frac{\pi }{2}-x\right)+\cos x\sin \left(\frac{\pi }{2}-y\right)=0$
$\Rightarrow$ $\cos y\sin x-\sin y\cos x+\sin y\sin x$
$+\cos x\cos y=0$
$\Rightarrow$ $(\sin x\cos y-\cos x\sin y)+(\cos x\cos y$
$+\sin x\sin y)=0$
$\Rightarrow$ $\sin (x-y)+\cos (x-y)=0$
$\Rightarrow$ $\sin (x-y)=-\cos (x-y)$
$\Rightarrow$ $\tan (x-y)=-1$
$\Rightarrow$ $x-y=n\pi -\frac{\pi }{4}\Rightarrow x=n\pi -\frac{\pi }{4}+y$