Is tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) ?

The answer to whether tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) is True

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prove tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x))

prove

tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

True

Solution steps

tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

Manipulating left side

tan (x + \frac{π}{3})

Rewrite using trig identities

= \frac{\frac{1}{2} sin ( x ) + \frac{3}{2} cos ( x )}{\frac{1}{2} cos ( x ) - \frac{3}{2} sin ( x )}

Simplify

\frac{cos ( x ) \frac{3}{2} + \frac{1}{2} sin ( x )}{cos ( x ) \frac{1}{2} - \frac{3}{2} sin ( x )} : \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

= \frac{sin ( x ) + cos ( x ) 3}{cos ( x ) - sin ( x ) 3}

Manipulating right side

\frac{tan ( x ) + 3}{1 - 3 tan ( x )}

Express with sin, cos

= \frac{sin ( x ) + cos ( x ) 3}{cos ( x ) - sin ( x ) 3}

We showed that the two sides could take the same form

\Rightarrow True

csc (x^{4}) - cot (x^{4}) = 2 csc (x) - 1

\frac{csc ( θ ) - sin ( θ )}{cot ( θ )} = cos (θ)

\frac{1}{sin ( x )} = arcsin (x)

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

cot (x) = arctan (x)

Is tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) ?
The answer to whether tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) is True