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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Simplify

- \frac{1}{1 + tan ( x )} + \frac{1}{1 - tan ( x )} : \frac{2 tan ( x )}{( tan ( x ) + 1 ) ( - tan ( x ) + 1 )}

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

Rewrite using trig identities

= tan (2 x)

We showed that the two sides could take the same form

\Rightarrow True

sin (θ + \frac{π}{2}) = cos (θ)

sin (2 x) = \frac{2 cot ( x )}{csc ^{2} ( x )}

(1 - cos (x)) (1 + sec (x)) = sec (x) - cos (x)

6 sin (5 π - x) = 6 sin (x)

2 sin (θ) cos (θ) = sin (2 θ)

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