Question

Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x

Answer 1

Let y = `cot^-1((cos x)/(1 + sinx))`

= `tan^-1((1 + sinx)/(cos x))`

= `tan^-1[(cos^2(x/2) + sin^2(x/2) + 2sin(x/2)cos(x/2))/(cos^2(x/2) - sin^2(x/2))]`

= `tan^-1[{cos(x/2) + sin(x/2)}^2/([cos(x/2) + sin(x/2)][cos(x/2) - sin(x/2)])]`

= `tan^-1[(cos(x/2) + sin(x/2))/(cos(x/2) - sin(x/2))]`

= `tan^-1[(1 + tan(x/2))/(1 - tan(x/2))]`

= `tan^-1[(tan(pi/4) + tan(pi/2))/(1 - tan(pi/4)tan(x/2))]`

= `tan^-1[tan(pi/4 + x/2)]`

∴ y = `pi/4 + x/2`

Differentiating w. r. t. x, we get

`("d"y)/("d"x) = "d"/("d"x)(pi/4 + x/2) = 0 + 1/2 = 1/2`