How do I prove these trig identities?

Question

secx-cosx/tanx =sinx

And

1+sinx/cosx+ cox/ 1+sinx=2secx

Answer 1

I will show you the first problem to give you an idea how to approach proving trig identities.

Here are a few basic identities that will help:

sinx/cosx = tanx
secx = 1/cosx
cos^2x + sin^2x = 1

Let's put everything into sine and cosine form and work the left hand side:

(secx - cosx)/tanx = sinx
(1/cosx - cosx)/(sinx/cosx) = sinx
(1/cosx - cosx)(cosx)/sinx = sinx
(cosx/cosx - cos^2x)/sinx = sinx
(1 - cos^2x)/sinx = sinx
sin^2x/sinx = sinx
sinx = sinx

And there you have it!