Answer to Question #1850 in Abstract Algebra for sally

Question

Verify the identity
1+sinx/cosx+cosx/1+sinx=2secx

Answer 1

(1+sinx)/cosx+cosx/(1+sinx)=2secx

Let's consider the left part of the expression:
((1+sinx)² + cos²x)/(cosx(1 +sinx)) = (1+2sinx +sin²x +cos²x)/(cosx*(1+sinx)) =

= (2 + 2 sinx)/(cosx*(1+sinx)) = 2(1+sinx)/(cosx (1+sinx)) = 2/cosx = 2 secx
2secx = 2secx

The identity is proved.