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Question
Differentiate the following w.r.t.x:
tan–1 (cosec x + cot x)
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Solution
Let y = tan–1 (cosec x + cot x)
= `tan^-1(1/sinx + cosx/sinx)`
= `tan^-1((1 + cosx)/(sinx))`
= `tan^-1[(2cos^2(x/2))/(2sin(x/2)*cos(x/2))]`
= `tan^-1[cot(x/2)]`
= `tan^-1[tan(π/2 - x/2)]`
= `π/2 - x/2`
Differentiating w.r.t.x, we get
`dy/dx = d/dx(π/2 - x/2)`
= `d/dx(π/2) - 1/2 d/dx(x)`
= `0 - 1/2 xx 1`
= `-1/2`
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