What is the formula for tan(a + b)?

The formula for the tangent of the sum of two angles, ( \tan(a + b) ), is derived from the definitions of tangent in relation to sine and cosine. Specifically, the tangent of an angle is defined as the ratio of the sine and cosine of that angle.

The tangent addition formula states that:

[

\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}

]

This formula comes from the identity for sine and cosine:

[

\tan(a) = \frac{\sin(a)}{\cos(a)} \quad \text{and} \quad \tan(b) = \frac{\sin(b)}{\cos(b)}

]

When we apply the angle addition formulas for sine and cosine:

[

\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)

]

[

\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)

]

We can express ( \tan(a + b) ) as follows:

[

\tan(a + b) = \frac{\sin(a + b)}{\cos(a + b