Prove the following identities:

Question

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

*Sorry for any confusing parenthesis.*

My work:
I simplified the left side to
a. ((sinx) / (cosx)) + (1 / cosx) , then
b. (sinx + 1) / cosx = (cos(x)) / (1-sin(x))

I don't know how to verify the rest.

Answer 1

good start

LS = sinx/cosx + 1/cosx
= (sinx + 1)/cosx
multiply top and bottom by (sinx - 1)/(sinx - 1)
= (sinx + 1)/cosx * (sinx - 1)/(sinx - 1)
= (sin^2 x - 1)/(cosx(sinx - 1))
= cos^2 x/(cosx(sinx - 1))
= cosx/(sinx - 1)
= RS

Answer 2

Thank you so much! I understand it now.