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Question

Prove that
tan1(6x8x3112x2)tan1(4x14x2)=tan12x;|2x|<13.

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Solution

L.H.S = tan1(6x8x3112x2)tan1(4x14x2)

=tan1[3×2x(2x)313×(2x)2]tan1[2×2x1(2x)2]

Considering tan12x=θ then 2x=tanθ

Now,

= tan1[3tan θ(tan θ)313(tan θ)2]tan1[2tan θ1tan2 θ]

= tan1[3tan θtan3 θ13 tan2θ]tan1[2tan θ1tan2 θ]

= tan1(tan 3θ)tan1(tan 2θ)

= 3θ2θ=θ=tan12x = R.H.S.

Hence proved.


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