To prove the identity sin^4(α) + cos^4(α) * cot(2α) = cot(2α), we need to simplify the left side of the equation using various trigonometric identities.

Starting with the left side of the equation:
sin^4(α) + cos^4(α) cot(2α)
= sin^2(α)^2 + cos^4(α) (cos(2α) / sin(2α))
= sin^2(α)^2 + cos^4(α) ((cos^2(α) - sin^2(α)) / (2 sin(α) cos(α)))
= sin^2(α)^2 + cos^4(α) (cos^2(α) - sin^2(α)) / (2 sin(α) cos(α))
= sin^2(α)^2 + cos^4(α) cos^2(α) - cos^4(α) sin^2(α) / (2 sin(α) cos(α))
= sin^2(α)^2 + cos^4(α) cos^2(α) - sin^2(α)^2 / (2 sin(α) cos(α))
= sin^2(α)^2 + cos^4(α) cos^2(α) - sin^2(α)^2 / (2 sin(α) cos(α))
= sin^2(α)^2 + cos^2(α)^2 - sin^2(α)^2 / (2 sin(α) cos(α))
= cos^2(α)^2 / (2 sin(α) cos(α))
= cos^2(α) / (2 * sin(α))
= cot(α) / 2
= (1 / tan(α)) / 2
= cot(α) / 2
= cot(2α)

Therefore, sin^4(α) + cos^4(α) * cot(2α) simplifies to cot(2α), proving the given identity.