Question

How would you prove this trigonomic identity? Please show steps.


`( \cos( \alpha ) )/(1 + \sin( \alpha ) ) + ( \cos( \alpha ) )/(1 - \sin( \alpha ) ) = 2 \sec( \alpha )`
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Answer 1

Start by combining the fractions:

`(\cos\alpha)/(1+\sin\alpha)\cdot(1-\sin\alpha)/(1-\sin\alpha)+(\cos\alpha)/(1-\sin\alpha)\cdot(1+\sin\alpha)/(1+\sin\alpha)`

`(\cos\alpha(1-\sin\alpha)+\cos\alpha(1+\sin\alpha))/((1+\sin\alpha)(1-\sin\alpha))`

`(2\cos\alpha)/(1-\sin^2\alpha)`

Recall the Pythagorean identity:

`(2\cos\alpha)/(\cos^2\alpha)`

Then cancel a factor of
`\cos\alpha` and use the definition of secant:

`\frac2{\cos\alpha}=\boxed{2\sec\alpha}`