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