The tan 2x formula is a double angle trigonometric identity that expresses tan(2x) in terms of tan(x). It is derived from the sine and cosine double angle formulas.
Formula: tan 2x = 2 tan x / (1 − tan²x).
Undefined When: tan²x = 1, i.e., when tan x = ±1 (x = 45° or 135°).
Related: sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x.
Also Written As: tan 2θ = 2 tan θ / (1 − tan²θ).
tan 2x = 2 tan x / (1 − tan²x)
This is the standard and most commonly used form.
We know:
Now: tan 2x = sin 2x / cos 2x = (2 sin x cos x) / (cos²x − sin²x)
Divide both numerator and denominator by cos²x: = (2 sin x / cos x) / (1 − sin²x/cos²x) = 2 tan x / (1 − tan²x)
If tan x = 1/2, find tan 2x. tan 2x = 2(1/2) / (1 − (1/2)²) = 1 / (1 − 1/4) = 1 / (3/4) = 4/3
tan 2x = 2 tan x / (1 − tan²x).
tan 2x is undefined when tan²x = 1, i.e., when tan x = ±1 or x = 45°, 135°, etc. (making the denominator zero).
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