Prove that:`cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2, x in (0,pi/4)`

Explanation

Let `y="cot"^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))]`
Now `sqrt(1+sinx)=sqrt("sin"^(2)(x)/(2) +"cos"^(2)(x)/(2)+2"sin"(x)/(2)"cos"(x)/(2))`
`=sqrt(("sin"(x)/(2)+"cos"(x)/(2))^(2))="sin"(x)/(2)+"cos"(x)/(2)`
and `sqrt(1-sinx)=sqrt("sin"^(2)(x)/(2) +"cos"^(2)(x)/(2)-2"sin"(x)/(2)"cos"(x)/(2))`
`=sqrt(("cos"(x)/(2)-"sin"(x)/(2))^(2))="cos"(x)/(2)-"sin"(x)/(2)`
`" " ( :' 0 lt x lt (pi)/(2))`
` :. y="cot"^(-1)[(("sin"(x)/(2)+"cos"(x)/(2))+("cos"(x)/(2)-"sin"(x)/(2)))/(("sin"(x)/(2)+"cos"(x)/(2))-("cos"(x)/(2)-"sin"(x)/(2)))]`
`="cot"^(-1)((2"cos"(x)/(2))/(2"sin"(x)/(2)))="cot"^(-1)("cot"(x)/(2))=(x)/(2)`
`=(dy)/(dx)=(d)/(dx)((x)/(2))=(1)/(2)`