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Question
Prove the following trigonometric identities.
`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`
Solution
We need to prove `sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`
Here, rationaliaing the L.H.S, we get
`sqrt((1 - cos A)/(1 + cos A)) = sqrt((1 - cos A)/(1 +cos A)) xx sqrt((1 - cos A)/(1 - cos A))`
`= sqrt((1 - cos A)^2/(1 - cos^2 A))`
Further using the property, `sin^2 theta + cos^2 theta = 1` we get
So,
`sqrt((1 - cos A)^2/(1 - cos^2 A)) = sqrt((1 - cos A)^2/sin^2 A`
`= (1 - cos A)/sin A`
`= 1/sin A - cos A/sin A`
= cosec A - cot A
Hence proved.
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We have, 1 + cot2θ = cosec2θ
1 + `square` = cosec2θ
1 + `square` = cosec2θ
`(square + square)/square` = cosec2θ
`square/square` = cosec2θ ......[Taking root on the both side]
cosec θ = `41/9`
and sin θ = `1/("cosec" θ)`
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∴ sin θ = `9/41`
The value is cosec θ = `41/9`, and sin θ = `9/41`
