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Prove the Following Trigonometric Identities. (Cot^2 A(Sec a - 1))/(1 + Sin A) = Sec^2 a ((1 - Sin A)/(1 + Sec A)) - Mathematics

Prove the following trigonometric identities.

`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`

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Solution

We have to prove `(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`

We know that `sin^2 A = cos^2 A = 1`

`So,

`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 = sec A))`

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

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

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

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

`= (cos A (1 - cos A))/((1 - cos A)(1 + cos A)(1 + sin A))`

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

`= (1/sec A)/((1 + 1/sec A)(1 + sin A))`

`= (1/sec A)/(((sec A + 1)/sec A)) (1 + sin A)`

`= 1/((sec A +1)(1 + sin A))`

Multiplying both the numerator and denominator by (1 - sin A), we have

`= (1 - sin A)/((sec A + 1)(1 + sin A)(1 - sin A))`

`= (1 - sin A)/((sec A + 1)(1 - sin^2 A))`

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

`= sec^2 A ((1 - sin A))/((sec A + 1))`

`= sec^2 A ((1 - sin A)/(1 + sec A))`

Hence proved.

 

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APPEARS IN

RD Sharma Class 10 Maths
Chapter 11 Trigonometric Identities
Exercise 11.1 | Q 67 | Page 46
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