Trig Identities With Higher Exponents?

Question

How do I prove or establish trig identities with exponents higher than 2?

SAMPLES:

(1) tan3(x) = (3tan(x) - tan^3 (x))/(1 - 3(tan^2 (x))

(2) tan4(x) = (4tan(x) - tan^4 (x))/(1 - 4tan^3 (x))

Answer 1

if u are asking for higher exponents means u might be knowing the values of Tan2x
for our probelem tan3x=tan(2x+x)
and for tan4x=tan(2x+2x)

Tan(x+y)=(tanx+tany)/(1-tanxtany)
for tan2x we have x=y in the above formula.
and for tan3x we have x=2x and y=x.
and go on computing with these identities.

Answer 2

hint:
tan3x=tan(x+2x)=?