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Question
`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec theta)`
Solution
LHS= `tan theta/((1-cot theta))+ cot theta/((1-tan theta))`
=`tan theta/((1-cos theta/sin theta)) + cot theta/((1-sin theta/cos theta))`
=`(sin theta tan theta)/((sin theta- cos theta))+(cos theta cot theta)/((cos theta - sin theta))`
=`(sin theta xx (sin theta) / (cos theta) cos theta xx (cos theta) / (sin theta))/((sin theta - cos theta))`
=`((sin ^2 theta cos ^2 theta)/(cos theta sin theta))/((sin theta-cos theta))`
=`( sin ^3 theta - cos ^3 theta)/(cos theta sin theta (sin theta - cos theta))`
=` ((sin theta - cos theta)(sin ^2 theta + sin theta cos theta + cos ^2theta ))/(cos theta sin theta (sin theta- costheta))`
=`(1+ sin theta cos theta)/(cos theta sin theta)`
=`1/(cos theta sin theta)+(sin theta cos theta)/(cos theta sin theta)`
=`1/(cos theta sin theta)+ (sin theta cos theta)/(cos theta sin theta)`
=`sectheta cosec theta +1`
=`1+ sec theta cosec theta`
=RHS
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