If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p² — 1) = 2p.

Prove the following by using trigonometric identities:

tan²A — tan²B =

Prove the following by using trigonometric identities:

2(sin⁶θ + cos⁶θ) — 3(sin⁴θ + cos⁴θ) + 1 = 0

If tanθ + sin = a and tanθ — sinθ = b, then prove that a² — b² =

acosθ + bsinθ = p and asinθ— bcosθ = q, then prove that a² + b² = p² + q².