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Question
Differentiate the following w.r.t. x : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
Solution
Let y = `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
= `tan^-1[sqrt((2cos^2(x/2))/(2sin^2(x/2)))]`
= `tan^-1[cot(x/2)]`
= `tan^-1[tan(pi/2 - x/2)]`
= `pi/(2) - x/(2)`
Differentiate the following w.r.t. x :
`"dy"/"dx" = "d"/"dx"(pi/2 - x/2)`
= `"d"/"dx"(pi/2) - (1)/(2)"d"/"dx"(x)`
= `0 - (1)/(2) xx 1`
= `-(1)/(2)`
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