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Question

Prove that: tan1(1+x1x1+x+1x)=π412cos1x;12x1.

OR If tan1(x2x4)+tan1(x+2x+4)=π4, find the value of x.

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Solution

LHS: Let Y=tan1(1+x1x1+x+1x)

Put x=cos2θθ=12cos1x (i)

Let Y=tan1(1+cos2θ1cos2θ1+cos2θ+1cos2θ)=tan1(2cosθ2sinθ2cosθ+2sinθ)

=tan1(cosθsinθcosθ+sinθ)=tan1(1tanθ1+tanθ)=tan1(tanπ4tanθ1+tanπ4tanθ)

=tan1tan(π4θ)=π4θ

=π212cos1x [By (i)]

=RHS

OR

The given equation is tan1(x2x4)+tan1(x+2x+4)=π4

tan1(x2x4)=tan1(x+2x+4)

tan1(x2x4)=tan1(1x+2x+41+1×x+2x+4)=tan1(22(x+3))

x2x4=1x+3 x3+x6=x4

x2=2 x=±2.


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