Problem 69 - Precalculus Essentials

Question

Rewrite each expression as a simplified expression containing one term.s $$\cos (\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta$$

Step-by-Step Solution

Answer

The simplified expression is \( \cos\alpha \)

Step 1: Identify the co-function identities

First of all, recognize the use of sine and cosine identities. The cosine addition formula is \( \cos(A + B) = \cos A \cos B – \sin A \sin B \), and the sine addition formula is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). The given expression can be seen as a combination of these two formulas.

Step 2: Rearrange and substitute

Reverse the given expression to align with the formula. It then becomes \( \cos\beta \cos(\alpha + \beta) + \sin\beta \sin(\alpha + \beta) \). Now, substitute \( \cos(\alpha + \beta) \) and \( \sin(\alpha + \beta) \) with the co-function identities. This leads to \[ \cos\beta [\cos\alpha \cos\beta - \sin\alpha \sin\beta] + \sin\beta [\sin\alpha \cos\beta + \cos\alpha \sin\beta] \]

Step 3: Simplify the expression

Simplify this expression. After the multiplication, the result is \( \cos^{2}\beta \cos\alpha - \cos\beta \sin\alpha \sin\beta + \cos\beta \sin\alpha \sin\beta + \cos\alpha \sin^{2}\beta \). Observe that \( -\cos\beta \sin\alpha \sin\beta \) and \( \cos\beta \sin\alpha \sin\beta \) cancel each other out because they are same in magnitude but opposite in sign. As a result, the expression simplifies to \( \cos^{2}\beta \cos\alpha + \cos\alpha \sin^{2}\beta \).

Step 4: Factor out common terms

Factor out the common term \( \cos\alpha \) from the simplified equation to get \[ \cos\alpha (\cos^{2}\beta + \sin^{2}\beta) \]. This can be further simplified because \( \cos^{2}\beta + \sin^{2}\beta = 1 \) (known as the Pythagorean trigonometric identity). The final simplified expression is \( \cos\alpha \).