Question

How do you prove `(cosx/(1+sinx))+((1+sinx)/cosx)=2secx`?

Answer 1

Convert the left side into terms with common denominator and add (converting `cos^2+sin^2` to `1` along the way); simplify and refer to definition of `sec = 1/cos`

Explanation:

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

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

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

`= 2/cos(x)`

`= 2* 1/cos(x)`

`= 2sec(x)`