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Question
Prove the following identities:
`("cosec" theta + cot theta)/("cosec" theta - cot theta) = ("cosec" theta + cot theta )^2 = 1 + 2 cot^2 theta + 2 "cosec" theta cot theta`
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Solution
Here, `( cosec theta + cot theta )/( cosec theta - cot theta)`
= `((cosec theta + cot theta) ( cosec theta + cot theta ))/(( cosec theta - cot theta ) ( cosec theta + cot theta))`
=` ((cosec theta + cot theta)^2)/(( cosec ^2 theta - cot^2 theta))`
=`((cosec theta + cot theta )^2) /1`
=`(cosec theta + cot theta )^2`
Again , `( cosec theta + cot theta )^2`
= ` cosec^2 theta + cot^2 theta + 2 cosec theta cot theta `
=` 1+cot^2 theta + cot^2 theta + 2 cosec theta cot theta (∵ cosec^2 theta - cot^2 theta =1)`
=` 1+2 cot^2 theta + 2 cosec theta cot theta `
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