Prove that:`(i)tan^(-1){(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))}=pi/4+x/2`,

Prove that:`(i)tan^(-1){(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt

1.	Prove that:`(i)tan^(-1){(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))}=pi/4+x/2`,
Answer» `cos 2 theta = 2 cos^2 theta - 1` `1 + cos 2 theta = 2 cos^2 theta` so, `1 + cos x = 2cos^2 (x/2)` Also `cos 2 theta = 1- 2sin^2 theta` `1- cos 2 theta = 2 sin^2 theta` so, `1 - cosx = 2sin^2(x/2)` now, putting it in the equation given `tan^-1{(sqrt(2cos^2(x/2)) - sqrt(2sin^2(x/2)))/(sqrt(2cos^2(x/2)) - sqrt(2sin^2(x/2)))}` `= tan^-1{ (sqrt2 cos(x/2) + sqrt2 sin(x/2))/(sqrt2 cos(x/2) - sqrt2sin(x/2))}` `= tan^-1{(1+tan(x/2))/(1- tan(x/2))}` `= tan^-1 {(tan (pi/2) + tan(x/2))/(1 - tan(pi/4) tan(x/2))}` `= tan^-1{tan(pi/4 + x/2)}` `= pi/4 + x/2 ` = RHS hence proved

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In a ` A B C ,`, if C is a right angle, then `tan^(-1)(a/(b+c))+tan^(-1)(b/(c+a))=``pi/3`(b) `pi/4`(c) `(5pi)/2`(d) `pi/6`
Prove that:`(i)tan^(-1){(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))}=pi/4+x/2`,
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