Is (cos(2x))/(1-sin(2x))=(1+tan(x))/(1-tan(x)) ?

The answer to whether (cos(2x))/(1-sin(2x))=(1+tan(x))/(1-tan(x)) is True

Popular >

prove (cos(2x))/(1-sin(2x))=(1+tan(x))/(1-tan(x))

prove

\frac{cos ( 2 x )}{1 - sin ( 2 x )} = \frac{1 + tan ( x )}{1 - tan ( x )}

True

Solution steps

\frac{cos ( 2 x )}{1 - sin ( 2 x )} = \frac{1 + tan ( x )}{1 - tan ( x )}

Manipulating left side

\frac{cos ( 2 x )}{1 - sin ( 2 x )}

Rewrite using trig identities

= \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

Manipulating right side

\frac{1 + tan ( x )}{1 - tan ( x )}

Express with sin, cos

= \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )}

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 - sin ^{2} ( x )}{1 + cos ( x )} = cos (x)

2 sin^{2} (x) - 1 = 1 - 2 cos^{2} (x)

(tan (x) + cot (x)) (sin (x) cos (x)) = 1

sin (2 x) = sin^{2} (x) - cos^{2} (x)

cos (x) - sin^{2} (x) cos (x) = cos^{3} (x)

Is (cos(2x))/(1-sin(2x))=(1+tan(x))/(1-tan(x)) ?
The answer to whether (cos(2x))/(1-sin(2x))=(1+tan(x))/(1-tan(x)) is True