Formulas for the tangent function can be derived from similar formulas involving the sine and cosine. The sum identity for tangent is derived as follows:

To determine the **difference identity for tangent**, use the fact that tan(−β) = −tanβ.

**Example 1:** Find the exact value of tan 75°.

Because 75° = 45° + 30°

**Example 2:** Verify that tan (180° − *x*) = −tan *x*.

**Example 3:** Verify that tan (180° + *x*) = tan *x*.

**Example 4:** Verify that tan (360° − *x*) = − tan *x*.

The preceding three examples verify three formulas known as the **reduction identities for tangent**. These reduction formulas are useful in rewriting tangents of angles that are larger than 90° as functions of acute angles.

The **double‐angle identity for tangent** is obtained by using the sum identity for tangent.

The **half‐angle identity for tangent** can be written in three different forms.

In the first form, the sign is determined by the quadrant in which the angle α/2 is located.

**Example 5:** Verify the identity

**Example 6**: Verify the identity tan (α/2) = (1 − cos α)/sin α.

**Example 7:** Verify the identity tan (α − 2) = sin π/(1 + cos α).

Begin with the identity in Example 6.

**Example 8:** Use a half‐angle identity for the tangent to find the exact value for tan 15°.

What follows are two alternative solutions.