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Question

Prove that b2x2a2y2=a2b2, if:
i) x=asecθ,y=btanθ
ii) x=acosecθ,y=bcotθ

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Solution

b2x2a2y2=a2b2
i) x=asecθ,y=btanθ
LHS=b2x2a2y2
=b2a2sec2θa2b2tan2θ
=a2b2(sec2θtan2θ)
=a2b2=RHS
sin2+cos2θ=1
1sin2θ=cos2θ
Dividing both sides by cos2θ
sec2θtan2θ=1
(sinθcosθ=tanθ)
ii) acscθ=x
bcotθ=y
LHS=b2x2a2y2
=b2a2csc2θa2b2cot2θ
=a2b2(csc2θcot2θ)
=a2b2
=RHS
csc2θ
cot2θ
=1

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