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

Question

2012 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-08-12T15:37:11

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

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

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