Question

Prove that : cos 2x cos 2 y + sin^2 (x - y) - sin^2 (x + y) = cos (2x + 2y).
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Answer 1

Answer:

Step-by-step explanation:

To prove: cos 2θ cos2φ + sin^2 (θ – φ) – sin^2(θ + φ) = cos (2θ + 2φ)

Proof:

LHS = cos 2θ cos2φ + sin2 (θ – φ) – sin2(θ + φ)

= cos 2θ cos2φ + sin (θ – φ + θ + φ) sin(θ – φ - θ - φ) [Using: sin2 A - sin2 B = sin(A + B) sin (A - B)]

= cos 2θ cos2φ + sin 2θ sin(-2φ)

= cos 2θ cos2φ - sin 2θ sin 2φ

= cos (2θ + 2φ) [using: cos (A + B) = cos A cos B - sin A sin B]

= RHS [Hence Proved]

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Answer 2

Answer:

Sin2x-sin2y/cos2y-cos2x

={2cos(2x+2y)/2sin(2x-2y)/2}/{2sin(2y+2x)/2sin(2x-2y)/2}

[∵, sinC-sinD=2cos(C+D)/2sin(C-D)/2 and cosC-cosD=2sin(C+D)/2sin(D-C)/2]

=cos(x+y)sin(x-y)/sin(x+y)sin(x-y)

=cot(x+y) (Proved