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Question
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
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Solution
Let y = `tan^-1 [sqrt((1 - cos x)/(1 + cos x))]`
= `tan^-1 [sqrt((2sin^2 x/2)/(2 cos^2 x/2))]` ......`[(because 1 - cos x = 2sinx^2 x/2),(1 + cos x = 2 cos^2 x/2)]`
= `tan^-1 [(sin x/2)/(cos x/2)]`
= `tan^-1 [tan x/2]`
∴ y = `x/2`
Differentiating both sides w.r.t. x
`"dy"/"dx" = 1/2 "d"/"dx"(x)`
= `1/2 * 1`
= `1/2`
Hence, `"dy"/"dx" = 1/2`
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