# How do i prove this identity? 1/1-sinA + 1/1+sinA = 2/cos^2A ?

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### 2 Answers

To prove this identity, combine the fractions on the left side by adding them. The common denominator is `(1 - sinA)(1+sinA) = 1 - sin^2A = cos^2A`

The first fraction with the new denominator will be

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

The second fraction with the new denominator will be

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

Then the sum of the two fractions on the left side will be

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

This is equal to the right side of the given identity, so the identity is proven.

Taking the common denominator on left hand side,

1/(1-sinA) + 1/(1+sinA) = (1+sinA + 1-sinA)/[(1-sinA)(1+sinA)] = 2/(1-sin^2A) = 2/cos^2A

(using the identity cos^2A = 1-sin^2A) .

Hence Proved.