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

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

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

prove

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

True

Solution steps

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

Manipulating left side

tan (2 x)

Rewrite using trig identities

tan (2 x)

Rewrite as

= tan (x + x)

Use the Angle Sum identity:

tan (s + s) = \frac{tan ( s ) + tan ( s )}{1 - tan ( s ) tan ( s )}

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

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

Simplify

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

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

Add similar elements:

tan (x) + tan (x) = 2 tan (x)

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

1 - tan (x) tan (x) = 1 - tan^{2} (x)

1 - tan (x) tan (x)

tan (x) tan (x) = tan^{2} (x)

tan (x) tan (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

tan (x) tan (x) = tan^{1 + 1} (x)

= tan^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= tan^{2} (x)

= 1 - tan^{2} (x)

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{cot ^{2} ( t )}{cos ( t )} = cot (t) csc (t)

\frac{sin ( x ) + tan ( x )}{1 + sec ( x )} = sec (x)

- csc^{2} (x) = - 1 - (cot (x))^{2}

\frac{1}{sin ^{2} ( x ) tan ( x )} = - sec (x)

\frac{1}{csc ( θ )} = - cos (θ)

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