Double Angle: Tangent

This identity tells you how to compute the tangent of an angle that is twice a known angle using only the tangent of the original angle. It comes from the tangent addition formula, since \(\tan(\theta+\theta)\) can be expanded and then simplified to produce the ratio \(\frac{2\tan\theta}{1-\tan^2\theta}\). The formula is especially useful because it turns a doubled-angle problem into a single tangent value, though it fails when \(1-\tan^2\theta=0\), where the tangent is undefined.

Use this when you know \(\tan\theta\) and need \(\tan 2\theta\) without finding the angle itself. It is common in trigonometric simplification, solving equations, and checking identities in pre-calculus and calculus.

• Forgetting to square \(\tan\theta\) in the denominator
• Using \(\frac{2\tan\theta}{1+\tan^2\theta}\) instead of the minus sign
• Ignoring values of \(\theta\) where the denominator becomes zero