Prove the following identities :
(asin θ + bcos θ)2 + (acos θ – bsin θ)2 = a2 + b2
Taking LHS = (asin θ + bcos θ)2 + (acos θ – bsin θ )2
Using the identity,(a + b)2 = (a2 + b2 + 2ab) and (a – b)2 = (a2 + b2 – 2ab)
= a2 sin2 θ + b2 cos2 θ + 2 ab sin θ cos θ + a2 cos2 θ + b2 sin2 θ – 2 ab sin θ cos θ
= a2 sin2 θ+ a2 cos2 θ + b2 sin2 θ + b2 cos2 θ
= a2 (sin2 θ + cos2 θ) + b2 (sin2 θ + cos2 θ)
= a2 + b2 [∵ cos2 θ + sin2 θ = 1]
=RHS
Hence Proved