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Question

If (tanθ+1cosθ)2+(tanθ1cosθ)2=m(1+sin2θ1sin2θ).Find m

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Solution

tanx=sinxcosx
So, LHS:-
Expand and put value
a2+b2=(a+b)22ab
a=tanθ+1cosθ,b=tanθ1cosθ
LHS=(tanθ+1cosθ+tanθ1cosθ)22(tanθ+1cosθ)(tanθ1cosθ)
(2tanθ)22(tanθ+1cosθ)(tanθ1cosθ)
4tan2θ2(tan2θ1cos2θ)
4tan2θ2tan2θ+2cos2θ
2tan2θ+2cos2θ
2sin2θcos2θ+2cos2θ
2(1+sin2θ)cos2θ sin2θ+cos2θ=1
2(1+sin2θ)1sin2θ
m=2.

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