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Question
Prove the following:
`(cosx + sinx)/(cosx - sinx) - (cosx - sinx)/(cosx + sinx)` = 2tan2x
Solution
L.H.S. = `(cosx + sinx)/(cosx - sinx) - (cosx - sinx)/(cosx + sinx)`
= `((cosx + sinx)^2 - (cosx - sinx)^2)/((cosx - sinx)(cosx + sinx)`
= `((cos^2x + sin^2x + 2sinx cosx) - (cos^2x + sin^2x - 2sinx cosx))/(cos^2x - sin^2x)`
= `(1 + 2sinx cosx - 1 + 2sinx cosx)/(cos^2x - sin^2x)`
= `(2(2sinx cosx))/(cos^2x - sin^2x)`
= `(2sin2x)/(cos2x)`
= 2tan 2x
= R.H.S.
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