Question 1202935
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One of the many <a href="https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf">trig identities</a> is
{{{tan(A+B) = (tan(A)+tan(B))/(1-tan(A)*tan(B))}}}
Let's plug in A = x and B = x


{{{tan(A+B) = (tan(A)+tan(B))/(1-tan(A)*tan(B))}}}


{{{tan(x+x) = (tan(x)+tan(x))/(1-tan(x)*tan(x))}}}


{{{tan(2x) = (2*tan(x))/(1 - tan^2(x))}}}
this will be useful in substitution steps later on.



Go back to
{{{tan(A+B) = (tan(A)+tan(B))/(1-tan(A)*tan(B))}}}


Let's plug in 
A = x
B = 2x
to get the following
{{{tan(A+B) = (tan(A)+tan(B))/(1-tan(A)*tan(B))}}}


{{{tan(x+2x) = (tan(x)+tan(2x))/(1-tan(x)*tan(2x))}}}


{{{tan(3x) = (tan(x)+tan(2x))/(1-tan(x)*tan(2x))}}}


This is one large fraction with 
numerator = tan(x)+tan(2x)
denominator = 1-tan(x)*tan(2x)


In other words,
{{{tan(3x) = P/Q}}}
where
{{{P = tan(x)+tan(2x)}}}
and
{{{Q = 1-tan(x)*tan(2x)}}}


Let's apply a substitution in the numerator
{{{P = tan(x)+tan(2x)}}}


{{{P = tan(x)+(2*tan(x))/(1 - tan^2(x))}}}


{{{P = tan(x)*((1 - tan^2(x))/(1 - tan^2(x)))+(2*tan(x))/(1 - tan^2(x))}}}


{{{P = (tan(x) - tan^3(x))/(1 - tan^2(x))+(2*tan(x))/(1 - tan^2(x))}}}


{{{P = (tan(x) - tan^3(x)+2*tan(x))/(1 - tan^2(x))}}}


{{{P = (3tan(x) - tan^3(x))/(1 - tan^2(x))}}}


Apply that same substitution with the denominator.


{{{Q = 1-tan(x)*tan(2x)}}}


{{{Q = 1-tan(x)*((2*tan(x))/(1 - tan^2(x)))}}}


{{{Q = 1-(2*tan^2(x))/(1 - tan^2(x))}}}


{{{Q = (1 - tan^2(x))/(1 - tan^2(x))-(2*tan^2(x))/(1 - tan^2(x))}}}


{{{Q = (1 - tan^2(x)-2*tan^2(x))/(1 - tan^2(x))}}}


{{{Q = (1 - 3*tan^2(x))/(1 - tan^2(x))}}}


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Recall that this identity
{{{tan(3x) = (tan(x)+tan(2x))/(1-tan(x)*tan(2x))}}}
is of the form {{{P/Q}}} 


where
{{{P = tan(x)+tan(2x)}}}
and
{{{Q = 1-tan(x)*tan(2x)}}}


but we found these equivalent forms
{{{P = (3tan(x) - tan^3(x))/(1 - tan^2(x))}}}
and
{{{Q = (1 - 3*tan^2(x))/(1 - tan^2(x))}}}


The 2nd versions of P and Q involve the denominator {{{1-tan^2(x)}}}. 
Those expressions cancel when dividing.


Therefore,
{{{tan(3x) = (3tan(x) - tan^3(x))/(1 - 3*tan^2(x))}}}


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For the next part of the problem, here are a few hints:
{{{tan(3x) = 0}}}


{{{(3tan(x) - tan^3(x))/(1 - 3*tan^2(x)) = 0}}}


{{{3tan(x) - tan^3(x) = 0}}}


{{{tan(x)(3 - tan^2(x)) = 0}}}


I'll let you take over from here.
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