Question

Prove the identity cos x/1 - sin x = sec x + tan x

Answer 1

Proved

Explanation:

Given

Prove that

`(cos x)/(1 - sin x) = sec x + tan x`

`(cos x)/(1 - sin x)`

Multiply the numerator and denominator by 1 + sinx

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

Combine both fractions to form 1

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

Expand the denominator using difference of two squares;

`i.e.\ (a - b)(a + b) = a^2 - b^2`

The expression becomes

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

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

From trigonometry;
`1 - sin^2x = cos^2x`

The expression becomes

`(cos x (1 + sin x))/((cos^2 x))`

Divide the numerator and the denominator by cos x

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

Split fraction

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

From trigonometry;
`(1)/(cos x) = sec x \ and\ (sin\ x)/(cos\ x) = tan\ x`

So;

`(1)/(cos x) + (sin x)/(cos x)` =
`sec x + tan x`