Question

Verify the identity. ( 1-sin x)/cosx=cosx/(1+sin x)

Answer 1

See below.

Explanation:

I'm going to use what I assume is the correct question.

`(1 - \sin x)/(\cos x) = (\cos x)/(1 + \sin x)`

`(\cos x)/(\cos x) * (1 - \sin x)/(\cos x) = (\cos x)/(1 + \sin x)`

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

Now use the identity

`sin^2 x + cos^2 x = 1`

`cos^2 x = 1 - sin^2 x`

We replace
`\cos^2 x` in the left denominator.

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

Factor the difference of squares in the left side denominator.

`(\cos x(1 - \sin x))/((1 + \sin x)(1 - \sin x)) = (\cos x)/(1 + \sin x)`

`(\cos x)/(1 + \sin x) = (\cos x)/(1 + \sin x)`