How do i prove

sin^2 x cos^2 y - cos^2 x sin^2 y ≡ sin^2 x - sin^2 y

see below

Step-by-step explanation:

prove: sin²x cos²y - cos²x sin²y ≡ sin²x - sin²y

we notice that the Right side only has sin² functions, hence we can start by trying to remove the cos² functions from the Left side.

Recall the identity: sin²α + cos²α = 1, can be rearranged to give us:

cos²α = 1 - sin²a.

If we apply this to the left side of our equation, then

cos²y = 1-sin²y, and cos²x = 1-sin²x

Substituting these into the left side of the equation:

sin²x cos²y - cos²x sin²y

= sin²x (1-sin²y) - (1-sin²x) sin²y

= sin²x - (sin²x sin²y) - sin²y + (sin²x sin²y)

= sin²x - sin²y

= Right Side of equation (Proven!)