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Question

y=tan1u1u2 & x=sec112u21, u(0,12) (12,1), prove that 2dydx+1=0.

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Solution

y=tan1u1u2 & x=sec112u21
Put u=sinθ in the above equations.
Then,
y=tan1sinθ1sin2θ & x=sec112sin2θ1
or, y=tan1sinθcosθ & x=sec11(cos2θ)
or, y=tan1tanθ & x=sec1(sec2θ)
or, y=θ & x=π2θ
Now, dydθ=1 & dxdθ=2
dydx=12
or, 2dydx+1=0.

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