Cosine of a Sum

This identity tells you how to find the cosine of a combined angle when that angle is written as a sum of two simpler angles. It comes from the geometry of rotations on the unit circle: rotating by A and then by B gives the same final direction as rotating by A+B, and the x-coordinate of that final point expands into the two cosine terms minus the sine-product term. The minus sign appears because the two vertical components interact in a way that reduces the final horizontal component.

Use this when you need the cosine of an angle that is awkward to evaluate directly but can be split into angles with known values, such as 15°, 75°, or 105°. It is also useful in trigonometric proofs, simplifying expressions, and deriving exact values without a calculator.

• Using plus instead of minus in the final term, or forgetting that the cosine sum identity subtracts sin A sin B
• Mixing up the angles in the sine and cosine factors, such as writing cos A sin B or sin A cos B in the wrong places
• Plugging in degrees correctly for the angle values but then evaluating the standard values like cos45° or sin30° incorrectly