What are the steps to proving the following Trigonometry identity?

Question

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

Answer 1

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

We know that secx = 1/cosx and tanx = sinx/cosx

Left hand side

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

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

Multiple by 1+sin(x)/1+sin(x)

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

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

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

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

= Right hand side