Q. 15 F4.7( 3 Votes )

Prove the following identities :

(asin θ + bcos θ)2 + (acos θ – bsin θ)2 = a2 + b2

Answer :

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


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