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Question
Prove the following identities:
cosec A(1 + cos A) (cosec A – cot A) = 1
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Solution
L.H.S. = cosec A(1 + cos A) (cosecA – cot A)
= `1/(sin A)(1 + cos A)(1/(sin A) - (cos A)/(sin A))`
`((1-cos A)/sin A)`
`1/sin A(1+cos A)((1-cos A)/sin A)`
`= ((1+ cos A)(1-cos A))/sin^2 A`
Apply the identity (1 + cosA) (1 − cosA) = 1 − cos2A
`= (1-cos^2A)/sin^2A`
`= sin^2A/sin^2A = 1`
cscA(1 + cosA) (cscA − cotA) = 1 proved
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