We can start by using the trigonometric identity:
sin (a - b) = sin a cos b - cos a sin b
and applying it to the first term in the expression:
2 sin (x - y) = 2 (sin x cos y - cos x sin y)
Then we can simplify the second term by using the identity:
cos (a + b) = cos a cos b - sin a sin b
cos (x + y) cos (x - y) = (cos x cos y - sin x sin y)(cos x cos y + sin x sin y)
= cos^2 x cos^2 y - sin^2 x sin^2 y
= cos^2 x/cos^2 y - sin^2 x/sin^2 y
= (cos^2 x - sin^2 x)/(cos^2 y sin^2 y)
= cos 2x/(sin^2 y cos^2 y)
Substituting these expressions back into the original equation, we get:
2 (sin x cos y - cos x sin y) * cos 2x/(sin^2 y cos^2 y) = cot x - cot y
Simplifying further by dividing both sides by 2sin(x - y)cos(x + y)cos(x - y), we obtain:
cos 2x/(sin^2 y cos^2 y) = (cot x - cot y)/(2 sin (x - y) cos (x + y) cos (x - y))
Therefore, the solution is cot x - cot y, which is equal to the right-hand side of the equation.