Differentiate the following functions with respect to `x` : `tan^(-1){(1+cosx)/(sinx)},0ltxltpi`

Explanation

y=`tan^(-1){(1+cosx)/(sinx)}`

`cosx=(1−tan^2(x/2))/(1+tan^2(x/2)​`

`sinx=(2tan(x/2))/(1−tan^2(x/2)​)​`

so,y=`tan^(-1)((1+(1−tan^2(x/2))/(1+tan^2(x/2)))/((2tan(x/2))/(1−tan2(x/2))​)​)`

=`tan^(-1)(1/tan(x/2))`

=`tan^(-1)(tan(pi/2-x/2))`

=`pi/2-x/2`

so,`dy/dx=-1/2`