What is the formula of tan 3x?

Answer: \(tan (3x) = \frac{(3tanx – tan^{3}x)}{(1 – 3tan^{2}x)}\)

We know that

\(\tan (A + B) = \frac{(tan A + tan B)}{(1-tan A tan B)}\)

So tan (3x) can be considered as tan (x + 2x)

\(\tan (x + 2x) = \frac{(tan x + tan 2x)}{(1-tan x tan 2x)}\)

tan (2x) = tan (x + x)

\(\tan (x + 2x) = \frac{(tan x + tan (x + x)}{(1 – tan x tan (x + x)}\) \(\tan (x + 2x) = \frac{tan x + \frac{(tan x + tan x)}{(1 – tan x tan x)}}{1 – tan x \frac{(tan x + tan x)}{(1 – tan x tan x)}}\) \(\tan (x + 2x) = \frac{tan x + \frac{2tanx}{1 – tan^{2}x}}{1 – \frac{2tan^{2}x}{1-tan^{2}x}}\) \(\tan (x + 2x) = \frac{\frac{tan x – tan^{3}x + 2tanx}{1-tan^{2}x}}{\frac{1-tan^{2}x – 2tan^{2}x}{1-tan^{2}x}}\)

\(\tan (x + 2x) =\frac{3tanx-tan^{3}x}{1-3tan^{2}x}\)

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