Expert: Sherman D. - 3/10/2008

Question
Hi Sherman, I hope you can help me.
Prove that:
(cosx/(1+sinx))+((1+sinx)/cosx)=2secx
I've tried many different ways, all unsuccessful. Thanks for your help!

Answer
(cos(x)/(1 + sin(x))) + ((1 + sin(x))/cos(x)) = 2sec(x)

(cos(x)^2 + (1 + sin(x))^2)/(cos(x) * (1 + sin(x)))
(cos(x)^2 + (1 + sin(x))(1 + sin(x)))/(cos(x) * (1 + sin(x)))
(cos(x)^2 + (1 + sin(x) + sin(x) + sin(x)^2))/(cos(x)(1 + sin(x)))
(cos(x)^2 + 1 + 2sin(x) + sin(x)^2)/(cos(x)(1 + sin(x)))
(cos(x)^2 + sin(x)^2 + 1 + 2sin(x))/(cos(x)(1 + sin(x)))
(1 + 1 + 2sin(x))/(cos(x)(1 + sin(x))
(2 + 2sin(x))/(cos(x)(1 + sin(x))
(2(1 + sin(x)))/(cos(x)(1 + sin(x)))
2/cos(x)
2sec(x)

http://math2.org/math/trig/identities.htm