Question

If $\sin x+\sin y=0=\cos x+\cos y$, then $\cos 2x+\cos 2y$ is equal to

• A. $-2\sin (x+y)$
• B. $-2\cos (x+y)$
• C. $2\sin (x+y)$
• D. $2\cos (x+y)$

Answer 1

The correct option is B $-2\cos (x+y)$

Consider,
${(\cos x+\cos y)}^2-{(\sin x+\sin y)}^2=0$
$\Rightarrow ({\cos }^{2}x+{\cos }^{2}y+2\cos x\cos y)-({\sin }^{2}x+{\sin }^{2}y+2\sin x\sin y)=0$
$\Rightarrow \cos 2x+\cos 2y=-2(\cos x\cos y-\sin x\sin y)$
$\Rightarrow \cos 2x+\cos 2y=-2\cos (x+y)$