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.