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Simplify tan^(-1).(cosx/(1-sinx))(-pi)/2...

Simplify `tan^(-1).(cosx/(1-sinx))(-pi)/2 lt x lt pi/2`

Text Solution

Verified by Experts

`tan^(-1)(cosx/(1-sinx))`
`=tan^(-1)((cos^2.x/2-sin^2.x/2)/((cos.x/2-sin.x/2)^2))`
`=tan^(-1)((cos.x/2+sin.x/2)/(cos.x/2-sin.x/2))`
`=tan^(-1) ((1+tan.x/2)/(1-tan.x/2))`
`=tan^(-1)(tan.(pi/4+x/2))`
`:' (-pi)/2 lt x lt pi/2`
`rArr (-pi)/4 lt x/2 lt pi/4`
`0 lt x/2+pi/4 lt pi/2`
`:. tan^(-1)(tan.(pi/4+x/2))=pi/4+x/2,-pi/2 lt x lt pi/2`
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