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cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sq...

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

Text Solution

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`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)`
`( :. x in (0,(pi)/(4))impliescos""(x)/(2) gt sin""(x)/(2))`
`=cos^(-1)""((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))`
`=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)=`RHS. Hence Proved.
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