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If pi<x<2pi, prove that (sqrt(1+cosx)+sq...

If `pi

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LHS`=(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))`
`=(sqrt(2)|cos""(x)/(2)|+sqrt(2)|sin""(x)/(2)|)/(sqrt(2)|cos""(x)/(2)|-sqrt(2)|sin""(x)/(2)|)=(|cos""(x)/(2)|+|sin""(x)/(2)|)/(|cos""(x)/(2)|-|sin""(x)/(2)|)`
`=(-cos""(x)/(2)+sin""(x)/(2))/(-cos""(x)/(2)-sin""(x)/(2))[because piltxlt2pi,therefore(pi)/(2)lt(x)/(x)ltpi]`
Dividing numerator and enominator by `sin(x//2)`, we get
`LHS=(cot""(x)/(2)-1)/(cot""(x)/(2)+1)=cot((x)/(2)+(pi)/(4))=RHS`.
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