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Prove the following identity, where the...

Prove the following identity, where the angles involved are acute angles for which the expressions are defined.
(iii) `(tantheta)/(1-cottheta)+(cottheta)/(1-tantheta)=1+sectheta cosectheta`

Text Solution

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Left hand side
=`(tantheta)/(1-cottheta)+(cottheta)/(1-tantheta)`
=`(tantheta)/(1-cottheta)+(cottheta)/(1-tantheta)`
=`(-tantheta)/(1-tantheta)+(cottheta)/(1-tantheta)`
=`(-tan^2theta+tantheta)/(1-tantheta)`
=`1/tantheta((1-tan^3theta)/(1-tantheta))`
=`1/tantheta(1+tan^2theta+tantheta)`
=`1+seccthetacosectheta`
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