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

Solution:

From the question it is given that, cos (x - y) - cos (x + y) = 2 sin x sin y.

As we know the sum and difference identities as,

cos (x - y) = cos x cos y + sin x sin y

cos (x + y) = cos x cos y - sin x sin y

Then, substitute the values in given identity,

cos (x - y) - cos (x + y) = (cos x cos y + sin x sin y) - (cos x cos y - sin x sin y)

cos (x - y) - cos (x + y) = cos x cos y + sin x sin y - cos x cos y + sin x sin y

By simplification we get,

cos (x - y) - cos (x + y) = sinx sin y + sin x sin y

cos (x - y) - cos (x + y) = 2 sinx siny

Therefore, the given identity cos (x - y) - cos (x + y) = 2 sin x sin y is verified.

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

Summary:

The identity cos (x - y) - cos (x + y) = 2 sin x sin y is verified.