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Prove that: (tan A)/(1 + sec A) - (tan A)/(1 - sec A) = 2 cosec A - Mathematics

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प्रश्न

Prove that:

`(tan A)/(1 + sec A) - (tan A)/(1 - sec A)` = 2 cosec A

प्रमेय
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उत्तर

LHS = `(tan A)/(1 + sec A) - (tan A)/(1 - sec A)`

= `tan A[1/(1 + sec A) - 1/(1 - sec A)]`

= `tan A[((1 - sec A) - (1 + sec A))/((1 + sec A) (1 - sec A))]`

= `tan A [(-2 sec A)/(1 - sec^2 A)]`

= `tan A [(-2 sec A)/(-tan^2 A)]`

= `(2 sec A)/(tan A)`

= `2 * 1/(cos A) * (cos A)/(sec A)`

= `2/(sin A)`

= 2 cosec A

= RHS

Hence proved.

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