Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(90° + θ)

Verified step by step guidance

1. Recall the cosine sum identity: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
2. Identify \(A = 90^\circ\) and \(B = \theta\) in the expression \(\cos(90^\circ + \theta)\).
3. Apply the identity: \(\cos(90^\circ + \theta) = \cos 90^\circ \cos \theta - \sin 90^\circ \sin \theta\).
4. Use the known values: \(\cos 90^\circ = 0\) and \(\sin 90^\circ = 1\), so substitute these into the expression.
5. Simplify the expression to write it as a trigonometric function of \(\theta\) alone.

Key Concepts

Cosine of a Sum Identity

The cosine of a sum identity states that cos(A + B) = cos A cos B - sin A sin B. This formula allows you to express the cosine of the sum of two angles in terms of the cosines and sines of the individual angles, which is essential for rewriting expressions like cos(90° + θ).

Special Angle Values

Certain angles such as 0°, 30°, 45°, 60°, and 90° have known sine and cosine values. For example, cos 90° = 0 and sin 90° = 1. Using these values simplifies expressions involving these angles, enabling the reduction of cos(90° + θ) to a function involving only θ.

Trigonometric Function Simplification

After applying identities, simplifying the resulting expression by substituting known values and combining like terms is crucial. This process helps rewrite complex trigonometric expressions into simpler forms involving a single variable, such as expressing cos(90° + θ) solely in terms of θ.