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Question
Find `bb(dy/dx)` in the following:
y = `sin^(-1) ((1-x^2)/(1+x^2))`, 0 < x < 1
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Solution
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
Putting x = tan θ
⇒ θ = tan−1 x
∴ y = `sin^-1 ((1 - tan^2 theta)/(1 + tan^2 theta))`
= sin−1 (cos 2 θ)
= `sin^-1 [sin (pi/2 - 2 theta)]`
= `pi/2 - 2theta`
= `pi/2 - 2 tan^-1 x`
On differentiating with respect to x,
`dy/dx = pi/2 d/dx (1) - 2 d/dx tan^-1 x`
`dy/dx = 0 - 2 xx 1/(1 + x^2)`
`dy/dx = -2/(1 + x^2)`
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