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Question
Prove the following identities:
`sintheta/(1 + costheta) + (1 + costheta)/sintheta` = 2cosecθ
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Solution
L.H.S. = `sintheta/(1 + costheta) + (1 + costheta)/sintheta`
= `(sin^2theta + (1 + cos theta)^2)/(sintheta*(1 + cos theta)`
= `(sin^2 theta + 1 + 2costheta + cos^2 theta)/(sintheta(1 + costheta))`
= `((sin^2 theta + cos^2theta) + 1 + 2costheta)/(sintheta(1 + costheta))`
= `(1 + 1 + 2costheta)/(sintheta(1 + costheta))`
= `(2 + 2costheta)/(sintheta(1 + costheta))`
= `(2(1 + costheta))/(sintheta(1 + costheta))`
= `2/sintheta`
= 2.cosec θ
= R.H.S.
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