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Prove each of the following identities :...

Prove each of the following identities :
`(i) ("cosec"theta + cot theta )/("cosec"theta - cot theta ) = ("cosec" theta + cot theta)^(2) = 1+2cot^(2) theta + 2"cosec" theta cot theta `
` (ii) (sec theta + tan theta ) /( sec theta - tan theta) =(sec theta + tan theta )^(2) = 1+ 2tan^(2) theta + 2 sec theta tan theta `

Text Solution

Verified by Experts

(i) `LHS =(("cosec" theta+cot theta))/(("cosec" theta-cot theta))xx(("cosec" theta+cot theta))/(("cosec" theta+cot theta))=("cosec" theta+cot theta)^(2).`
And, `("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. `
(ii) `LHS=((sec theta +tan theta))/((sec theta -tan theta))xx((sec theta+tan theta))/((sec theta+tan theta)).`
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Prove that: ("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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Knowledge Check

  • ( cosec theta - sin theta ) ( sec theta - cos theta ) ( tan theta + cot theta ) =

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    B
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    C
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    D
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