The formula for the tan2x formula tangent of a double angle often denoted as \( \tan(2x) \), relates the tangent of an angle to the tangent of twice that angle \( x \). Here’s how it’s derived:
Given the double-angle identity for tangent:
\[ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \]
This formula allows you to express \( \tan(2x) \) in terms of \( \tan(x) \), which can be useful in various trigonometric calculations and identities. We derive it from basic trigonometric identities, specifically those involving sine and cosine, expressed in terms of tangent.
The derivation starts with the double-angle identities for sine and cosine:
\[ \sin(2x) = 2\sin(x)\cos(x) \]
\[ \cos(2x) = \cos^2(x) - \sin^2(x) \]
We are expressing sine and cosine in terms of tangent \( (\tan(x) = \frac{\sin(x)}{\cos(x)}) \), we substitute these into the formulas and simplify to obtain the double-angle formula for tangent.
