Identify the x-values that are solutions for the equation.?

Question

How do you solve?

5√3tan(x+3) = 8√3tan(x)

A. x = (pi/6), (11pi/6)
B. x = (7pi/6) , (11pi/6)
C. x = (pi/6) , (7pi/6)
D. x = (5pi/6) , (11pi/6)

PLEASE HELP!

Answer 1

1) get rid of the √3 (divide both sides by √3)
(unclear if the tan belongs under the root; i'll continue as if not)
5 tan(x+3) = 8 tan(x)

Use:
tan(x+3) =
(tan(x)+tan(3)) / (1-tan(x)tan(3))
from Wikipedia

substitute:
5*(tan(x)+tan(3))/(1-tan(x)tan(3)) = 8* tan(x)

5tan(x) + 5tan(3) = 8tan(x) - 8tan(x)tan(x)tan(3).

5, 8 and tan(3) are constants

8tan(3)*(tan(x))^2 +13*tan(x) + 5tan(3)

From now on, you are solving a quadratic where the unknown is "tan(x)"

tan(x) =[ -13 +/- SQRT(169 - 160((tan(3)^2))] / (2*8tan(3))

etc.

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If the tangents were meant to be inside the square root, then square both sides:

75*tan(x+3) = 192*tan(x)

and proceed as above.