Prove the following: `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))] = x/2 ; x in(0,pi/4)`

Prove the following: `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+

1.	Prove the following: `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))] = x/2 ; x in(0,pi/4)`
Answer» `sqrt(1+sinx)=sqrt(sin^2(x/2)+cos^2(x/2)+2sin(x/2)cos(x/2)` `=cos(x/2)+sin(x/2)` `sqrt(1-sinx)=cos(x/2)-sin(x/2)` `cot^(-1)((cos(x/2)+sin(x/2)+cos(x/2)-sin(x/2))/(cos(x/2)+sin(x/2)-cos(x/2)+sin(x/2)))` `cot^(-1)cot(x/2)=x/2`.

`tan^(-1)[(cosx)/(1+sinx)]`is equal to`pi/4-x/2,forx in (-pi/2,(3pi)/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-(3pi)/2,pi/2)`
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