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Question

If y=tan(12cos11u21+u2+12sin12u1+u2) and x=2u1u2, then dydx

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Solution

Let us consider θϵ(π2,π2) such that u=tanθ.
Now, 1u21+u2=1tan2θ1+tan2θ=cos2θ
and, 2u1+u2=2tanθ1+tan2θ=sin2θ

Therefore, y=tan(12cos11u21+u2+12sin12u1+u2)
=tan(12cos11tan2θ1+tan2θ+12sin12tanθ1+tan2θ)
=tan(12cos1cos2θ+12sin1sin2θ) (Using formulae of cos2θ and sin2θ )
=tan(122θ+122θ)
=tanθ
y=u

Therefore dydx=1.


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