Question

How do you prove `cos (2x) / (1-sin 2x) = (1 + tan 2x) / (1-tan 2x)`?

Answer 1

`(cos 2x)/(1-sin 2x) ne (1+tan 2x)/(1-tan 2x)`

Explanation:

Assume that `(cos 2x)/(1-sin 2x) = (1+tan 2x)/(1-tan 2x)` is true for all `x`

Taking `2x = pi/4`

LHS: `cos(pi/4)/(1-sin(pi/4)) = sqrt(2)/(2-sqrt(2))`
RHS: `lim_(2x rarr pi/4) (1+tan(2x))/(1-tan(2x)) = oo`

Therefore `(cos 2x)/(1-sin 2x) ne (1+tan 2x)/(1-tan 2x)`