Question

Prove that x4 - y4 = b2 - a2, if a cot θ + b cosecθ = x2 and b cot θ + a cosec θ = y2

solve fast :|

Answer 1

We have x2 + y2 = (a sin θ + b cos θ)2 + (a cos θ – b sin θ)2

= (a2 sin2 θ + b2 cos2 θ + 2ab sin θ cos θ) + (a2cos2 θ + b2 sin2 θ - 2ab sin θ cos θ) 

= a2 sin2 θ + b2 cos2 θ + 2ab sin θ cos θ + a2 cos2 θ + b2 sin2 θ - 2ab sin θ cos θ

= a2 sin2 θ + b2 cos2 θ + a2 cos2 θ + b2 sin2 θ 

= a2 sin2 θ + a2 cos2 θ + b2 sin2 θ + b2 cos2 θ

= a2 (sin2 θ + cos2 θ) + b2 (sin2 θ + cos2 θ) 

= a2 (1) + b2 (1); [since, sin2 θ + cos2 θ = 1] 

= a2 + b2

Therefore, x2 + y2 = a2 + b2

which is the required θ-eliminate. 

Answer 2

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