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Question
Prove the following:
`cosx/(1 + sinx) = (cot(x/2) - 1)/(cot(x/2) + 1)`
Solution
L.H.S. = `cosx/(1 + sinx)`
= `(cos^2(x/2) - sin^2(x/2))/(cos^2(x/2) + sin^2(x/2) + 2sin(x/2)cos(x/2))`
= `([cos(x/2) - sin(x/2)][cos(x/2) + sin(x/2)])/[cos(x/2) + sin(x/2)]^2`
= `(cos(x/2) - sin(x/2))/(cos(x/2) + sin(x/2)`
= `(cos(x/2)/(sin(x/2)) - (sin(x/2))/(sin(x/2)))/(cos(x/2)/(sin(x/2)) + sin(x/2)/(sin(x/2))`
= `(cot(x/2) - 1)/(cot(x/2) + 1)`
= R.H.S.
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