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प्रश्न
Prove the following trigonometric identities.
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
उत्तर
We have to prove the following identity
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
Consider the LHS = `(1 + cos theta + sin theta)/(1 + cos theta - sin theta)`
`= ((1 + cos theta + sin theta)/(1 + cos theta - sin theta))((1 + cos theta + sin theta)/(1 + cos theta + sin theta))`
`= (1 + cos theta + sin theta)^2/((1 + cos theta)^2 sin^2 theta)`
`= (2 + 2(cos theta + sin theta + sin theta cos theta))/(2 cos^2 theta + 2 cos theta)`
`= (2(1 + cos theta)(1 + sin theta))/(2 cos theta (1 + cos theta))`
`= (1 + sin theta)/cos theta`
= RHS
Hence proved
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L.H.S = `square`
= `cos^2theta xx square .....[1 + tan^2theta = square]`
= `(cos theta xx square)^2`
= 12
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