The identity to be proved is: cos(theta) / 1 - sin(theta) = sec(theta) + tan(theta)

I'll rewrite this with x used instead of theta

We have to prove: cos x/ (1 - sin x) = sec x + tan x

Let's start from the right hand side

sec x + tan x

substitute sec x = 1/cos x and tan x = sin x / cos x

=> 1/ cos x + sin x / cos x

=> (1 + sin x)/cos x

multiply the numerator and denominator by (1 - sin x)

=> (1 + sin x)(1 - sin x)/(cos x)*(1 - sin x)

=> (1 - (sin x)^2)/ (cos x)*(1 - sin x)

=> (cos x)^2 / (cos x)*(1 - sin x)

=> cos x / (1 - sin x)

which is the left hand side

**This proves the identity cos(theta)/(1 - sin(theta)) = sec(theta) + tan(theta)**