यदि `y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))]`, जहाँ `0ltxlt(pi)/(2)` तो `(dy)/(dx)` किसके बराबर है ?

Options

A. `-1//2`
B. 2
C. `sinx+cosx`
D. `sinx-cosx`

Answer

A

Explanation

दिया है,
`y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))]`
`=cot^(-1)[(|"sin"(x)/(2)+"cos"(x)/(2)|+|"sin"(x)/(2)-"cos"(x)/(2)|)/(|"sin"(x)/(2)+"cos"(x)/(2)|-|-"sin"(x)/(2)-"cos"(x)/(2)|)]`
`because" "0ltxlt(pi)/(2)`
`because" "y=cot^(-1)`
`[(("sin"(x)/(2)+"cos"(x)/(2))+("sin"(x)/(2)-"cos"(x)/(2)))/(("sin"(x)/(2)+"cos"(x)/(2))-("sin"(x)/(2)-"cos"(x)/(2)))]`
`y=cot^(-1)("tan"(x)/(2))`
`=cot^(-1)[cot((pi)/(2)-(x)/(2))]=((pi)/(2)-(x)/(2))`
`rArr" "(dy)/(dx)=-(1)/(2)`