Using the sum formula for tangent, express \( \tan(a + b) \) in terms of \( \tan(a) \) and \( \tan(b) \).

The **sum formula for tangent** allows you to express \(\tan(a + b)\) in terms of \(\tan(a)\) and \(\tan(b)\). The formula is derived from the sine and cosine addition formulas and is given by: \[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \] ### Derivation: 1. **Start with the sine and cosine addition formulas:** \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] 2. **Express \(\tan(a + b)\) using sine and cosine:** \[ \tan(a + b) = \frac{\sin(a + b)}{\cos(a + b)} = \frac{\sin a \cos b + \cos a \sin b}{\cos a \cos b - \sin a \sin b} \] 3. **Divide the numerator and the denominator by \(\cos a \cos b\):** \[ \tan(a + b) = \frac{\frac{\sin a}{\cos a} + \frac{\sin b}{\cos b}}{1 - \frac{\sin a}{\cos a} \cdot \frac{\sin b}{\cos b}} = \frac{\tan a + \tan b}{1 - \tan a \tan b} \] ### Final Formula: \[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \] **Example Usage:** If you know that \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) and \(\tan 45^\circ = 1\), you can find \(\tan(30^\circ + 45^\circ) = \tan 75^\circ\) as follows: \[ \tan(75^\circ) = \frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}} \cdot 1} = \frac{1 + \sqrt{3}}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = 2 + \sqrt{3} \] Thus, \(\tan(75^\circ) = 2 + \sqrt{3}\).