What is the formula for tan(3x)?

The complete formula for tan(3x) is:

tan(3x) = (3tan x - tan³x) / (1 - 3tan²x)

(You can also write it using theta: tan(3θ) = (3tanθ - tan³θ) / (1 - 3tan²θ)).

Notice the pattern of the numbers 3 and 1:

• The numerator has the terms 3 and power of 3: (3tan x - tan³x)
• The denominator starts with 1 and ends with 3 and square: (1 - 3tan²x)
• The signs are always alternating: Minus on top, Minus on bottom.

To prove this formula, you use the standard addition formula: tan(A + B).

1. Write tan(3x) as tan(2x + x).
2. Apply the formula: tan(A+B) = (tan A + tan B) / (1 - tan A · tan B) tan(2x + x) = (tan(2x) + tan(x)) / (1 - tan(2x)tan(x))
3. Now, substitute the double angle formula for tan(2x), which is (2tan x) / (1 - tan²x).
4. When you substitute this fraction into the numerator and denominator, take the LCM, cross-multiply, and simplify. The complex terms will cancel out, leaving you with the final formula.