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Question
Prove the following identities:
`(1 + tan theta + cot theta)(sin theta - cos theta) = ((sec theta)/("cosec"^2 theta) - ("cosec" theta)/(sec^2 theta))`
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Solution
LHS = `(1 + tan theta + cot theta )(sintheta - cos theta) `
= `sin theta + tan theta sin theta + cot theta sin theta - cos theta - tan theta cos theta - cot theta cos theta `
= `sin theta + tan theta sin theta + cos theta/sin theta xx sin theta - cos theta - sin theta/cos theta xx cos theta - cot theta cos theta`
= `sin theta + tan theta sin theta + cos theta - cos theta - sin theta - cot theta cos theta`
= `tan theta sin theta - cot theta cos theta`
= `sin theta/cos theta xx 1/(cosec theta) - cos theta/sin theta xx 1/sec theta`
= `1/(cosec theta) xx 1/(cosec theta) xx sec theta - 1/sec theta xx 1/sec theta xx cosec theta`
= `sec theta/(cosec^2 theta) - (cosec theta)/sec^2 theta`
= RHS
Hence, LHS = RHS
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