Prove the following: tan4x = 4tanx(1-tan²x) / (1-6tan²x+tan⁴x)

tan4x = 2tan2x / (1-tan²2x) = 2(2tanx/(1-tan²x)) / (1-(2tanx/(1-tan²x))²) = 4tanx(1-tan²x) / (1-tan²x)² - 4tan²x = 4tanx(1-tan²x) / (1-2tan²x+tan⁴x) - 4tan²x = 4tanx(1-tan²x) / (1-2tan²x+tan⁴x) - 4tan²x(1-tan²x)²/(1-tan²x)² = 4tanx(1-tan²x) / (1-2tan²x+tan⁴x) - 4tan²x(1-2tan²x+tan⁴x)/(1-2tan²x+tan⁴x) = (4tanx(1-tan²x) - 4tan²x(1-2tan²x+tan⁴x)) / (1-2tan²x+tan⁴x) = (4tanx - 4tan³x - 4tan²x + 8tan⁴x - 4tan⁶x) / (1-2tan²x+tan⁴x) This approach is incorrect and leads to a more complex expression. Let's try a different approach.

tan4x = 2tan2x / (1 - tan²2x)
We know that tan2x = 2tanx / (1 - tan²x)
Substituting this into the expression for tan4x:
tan4x = 2(2tanx / (1 - tan²x)) / (1 - (2tanx / (1 - tan²x))²)
= (4tanx / (1 - tan²x)) / (( (1 - tan²x)² - 4tan²x) / (1 - tan²x)²)
= 4tanx(1 - tan²x) / (1 - 2tan²x + tan⁴x - 4tan²x)
= 4tanx(1 - tan²x) / (1 - 6tan²x + tan⁴x)
Hence proved