How Do You Derive the Double Angle Identity for Tan From the Compound Angle Identities?

The double angle identity is

$$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta},$$

and it is obtained by setting both angles equal in the compound angle identity for tangent. Equivalently, tangent addition can be derived by dividing the sine addition identity by the cosine addition identity.

Derivation

Start with the sine and cosine compound angle identities from AHL 3.10:

$$\sin(A+B)=\sin A\cos B+\cos A\sin B,$$ $$\cos(A+B)=\cos A\cos B-\sin A\sin B.$$

Since tangent is sine divided by cosine, form the ratio

$$\tan(A+B)=\frac{\sin A\cos B+\cos A\sin B}{\cos A\cos B-\sin A\sin B}.$$

Divide every term in the numerator and denominator by $\cos A\cos B$. This is the key algebraic step: every term must be divided by the same non-zero expression. The numerator becomes $\tan A+\tan B$, while the denominator becomes $1-\tan A\tan B$. Therefore,

$$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}.$$

Now set $A=B=\theta$. Since $A+B=2\theta$,

$$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}.$$

This substitution is legitimate because the compound identity holds for any angles where both sides are defined. Doubling $\theta$ means evaluating tangent at the combined angle, while algebraically the two tangent terms combine to give $2\tan\theta$ and their product becomes $\tan^2\theta$.

For example, if $\theta=\frac{\pi}{6}$, then $\tan\theta=\frac{1}{\sqrt{3}}$. Hence,

$$\frac{2\left(\frac{1}{\sqrt{3}}\right)}{1-\frac{1}{3}}=\sqrt{3}=\tan\left(\frac{\pi}{3}\right).$$

This verifies one case, but checking a single angle is not a proof of the identity.

Domain Restrictions

The derivation assumes $\cos A\cos B\neq 0$. The final expression also requires $1-\tan^2\theta\neq 0$.

When $\tan\theta=\pm 1$, the denominator is zero and $\tan(2\theta)$ is undefined. A common misconception is using a plus sign in the denominator. The minus sign comes from the cosine addition identity.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches HL, AHL 3.10: Compound angle identities. In a Paper 1 or Paper 2 show that question, state both compound identities, form the ratio, show the division by $\cos A\cos B$, and substitute $A=B=\theta$. Quoting only the final identity may not earn the method marks.

In a derivation, the command term show requires a connected chain of algebra rather than a numerical check. Keep equality signs aligned, mention the non-zero divisor, and finish with the required identity so the examiner can award each method step.