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Question
Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `
Solution
`cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))`
`=cot^(-1)((sqrt(cos^2(x/2)+sin^2(x/2)+2 sin(x/2)cos(x/2))+sqrt(cos^2(x/2)+sin^2(x/2)-2 sin(x/2)cos(x/2)))/(sqrt(cos^2(x/2)+sin^2(x/2)+2 sin(x/2)cos(x/2))-sqrt(cos^2(x/2)+sin^2(x/2)-2 sin(x/2)cos(x/2)))) [∵sin 2x=2 sin x cos x and sin^2 x+cos^2 x=1]`
`=cot^(-1)(sqrt((cos(x/2)+sin(x/2))^2+sqrt((cos(x/2)-sin(x/2))^2))/(sqrt((cos(x/2)+sin(x/2))^2)-sqrt((cos(x/2)-sin(x/2))^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) {((cos(x/2)+sin(x/2))+(cos(x/2)-sin(x/2)))/((cos(x/2)+sin(x/2))-(cos(x/2)-sin(x/2)))} [∵0<x<pi/4⇒cos(x/2)>sin (x/4)]`
`=cot^(-1)((2cos(x/2))/(2sin(x/2)))`
`=cot^(-1)(cotx/2)`
`=x/2`
`=RHS`
Hence proved
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