Question

Verify the identity. cos (x - y) - cos (x + y) = 2 sin x sin y

Answer 1

Let's start from the left side, and manipulate it to get the right side, which would prove the identity.

We know that:

i) $\displaystyle{ \cos(x-y)=\cos x \cos y+\sin x \sin y$

ii) $\displaystyle{ \cos(x+y)=\cos x \cos y-\sin x \sin y$.

From the above fundamental identities, we can rewrite the left hand side as:

$\displaystyle{ \cos x \cos y+\sin x \sin y-(\cos x \cos y-\sin x \sin y)=$


$\displaystyle{ \cos x \cos y+\sin x \sin y-\cos x \cos y+\sin x \sin y=2\sin x \sin y$,

which is what we needed to show.