Find the exact value of each expression. (Do not use a calculator.)
cos 105° (Hint: 105° = 60° + 45°)

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Recognize that the angle 105° can be expressed as the sum of two special angles: 105° = 60° + 45°, which allows us to use the cosine addition formula.

Recall the cosine addition formula: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).

Substitute \(A = 60^\circ\) and \(B = 45^\circ\) into the formula: \(\cos 105^\circ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ\).

Use the exact values of the trigonometric functions for 60° and 45°: \(\cos 60^\circ = \frac{1}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), and \(\sin 45^\circ = \frac{\sqrt{2}}{2}\).

Substitute these values into the expression and simplify step-by-step to find the exact value of \(\cos 105^\circ\).

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Angle Sum Identity for Cosine

The angle sum identity states that cos(A + B) = cos A cos B - sin A sin B. This formula allows us to find the cosine of a sum of two angles by using the cosines and sines of the individual angles, which is essential for evaluating cos 105° as cos(60° + 45°).

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Certain angles like 30°, 45°, 60°, and their multiples have well-known exact sine and cosine values. Knowing these values, such as cos 60° = 1/2 and sin 45° = √2/2, is crucial for calculating trigonometric expressions without a calculator.

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Simplification of Radicals

After applying the angle sum identity, the resulting expression often contains square roots. Being able to simplify radicals and combine like terms helps in expressing the final answer in its simplest exact form, such as simplifying expressions involving √2.

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