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Prove the Following Trigonometric Identities. (Cos Theta - Sin Theta + 1)/(Cos Theta + Sin Theta - 1) = Cosec Theta + Cot Theta - CBSE Class 10 - Mathematics

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Question

Prove the following trigonometric identities.

`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta  + cot theta`

Solution

We have to prove the following identity

`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta  + cot theta`

Consider the LHS = `(cos theta - sin theta + 1)/(cos theta + sin theta - 1)`

`= (cos theta - sin theta  +  1)/(cos theta +  sin theta  - 1) xx (cos theta +  sin theta +  1)/(cos theta +  sin theta + 1)`

`= ((cos theta + 1)^2 - (sin theta)^2)/((cos theta +  sin theta)^2 - (1)^2)`

`= (cos^2 theta +  1 +  2 cos theta - sin^2 theta)/(cos^2 theta +  sin^2 theta + 2 cos theta sin theta - 1)`

`= (cos^2 theta + 1 +  2 cos theta - (1 -  cos^2 theta))/(1 +  2 cos theta sin theta  - 1)`

`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`

`= (2 cos^2 theta +  2 cos theta)/(2 cos theta sin theta)`

`= (2 cos theta(cos theta + 1))/(2 cos theta sin theta)`

`= (cos theta + 1)/sin theta`

`= cos theta/sin theta + 1/sin theta`

`= cot theta + cosec theta`

= RHS

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Solution Prove the Following Trigonometric Identities. (Cos Theta - Sin Theta + 1)/(Cos Theta + Sin Theta - 1) = Cosec Theta + Cot Theta Concept: Trigonometric Identities.
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