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Question

If secθ+tanθ=x, show that x21x2+1=sinθ

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Solution

Given:- x=secθ+tanθ
To prove:- x21x2+1=sinθ
Proof:-
x=secθ+tanθ
x=1+sinθcosθ
Squaring both sides, we get
x2=(1+sinθ)2cos2θ
x2=(1+sinθ)21sin2θ
x2=(1+sinθ)2(1+sinθ)(1sinθ)
x2=(1+sinθ)1sinθ
Therefore,
x21x2+1
=(1+sinθ)1sinθ1(1+sinθ)1sinθ+1
=1+sinθ1+sinθ1+sinθ+1sinθ
=2sinθ2=sinθ
Hence proved.

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