Tangent of a Sum

This identity tells you how to compute the tangent of a combined angle from the tangents of two separate angles. It comes from writing tan as sin/cos, using the sine and cosine addition formulas for A+B, and then simplifying the resulting fraction. The denominator 1 - tan A tan B is important because it shows the formula breaks down whenever the combined angle is 90° plus an integer multiple of 180°, where tangent is undefined.

Use this when you know tan A and tan B separately but need tan(A+B) without first finding the angles themselves. It is especially useful in trigonometric proofs, exact-value calculations, and simplifying expressions in calculus, physics, and engineering.

• Using plus in the denominator instead of 1 - tan A tan B
• Forgetting that tangent is undefined when the denominator becomes 0
• Mixing up the formula for tangent sum with the formulas for sine or cosine addition