Prove: tan⁡(π+4x)=(4 tan⁡x-4 tan^3⁡ x)/(tan^4 ⁡x-6 tan^2 ⁡x+1)

Answer 1

tan⁡(π+4x) = (tanπ + tan4x)/(1 - tanπ tan4x) = tan4x
tan2x = 2tanx / (1-tan^2 x)
tan4x = 2tan2x/(1-tan^2 2x) = 4tanx/[1-tan^2 x)(1 - tan^2 2x)]

see what you can do with that