Question

\((\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}\) is (a) \(4 \cos ^{2}\left(\frac{\alpha+\beta}{2}\right)\) (b) \(4 \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)\) (c) \(-4 \sin ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\beta}{2}\) (d) \(4 \sin \frac{\alpha}{2} \cos \frac{\beta}{2}\)

Short answer

Question: Simplify the expression \((\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}\) and choose the correct option. a) \(2 + 2\cos(\alpha + \beta)\) b) \(4\cos^2\left(\frac{\alpha-\beta}{2}\right)\) c) \(2 + 2\cos(\alpha - \beta)\) d) \(1 + \cos(\alpha - \beta)\) Answer: (b) \(4\cos^2\left(\frac{\alpha-\beta}{2}\right)\)

Step-by-step solution

Utilize the square of a sum formula

We begin by expanding the squares in the given expression: \((\cos \alpha+\cos \beta)^2 = \cos^2 \alpha + 2 \cos \alpha \cos \beta + \cos^2 \beta\) and \((\sin \alpha+\sin \beta)^2 = \sin^2 \alpha + 2 \sin \alpha \sin \beta + \sin^2 \beta.\)

Combine the terms of the expressions

Sum the expressions from Step 1: \(\cos^2 \alpha + \cos^2 \beta + \sin^2 \alpha +\sin^2 \beta + 2\cos \alpha \cos \beta + 2\sin \alpha \sin \beta\)

Make use of the Pythagorean identity

Recall that \(\cos^2 x + \sin^2 x = 1\). Hence, by the Pythagorean identity, we can rewrite the expression in Step 2 as: \(1 + 1 + 2(\cos \alpha \cos \beta + \sin \alpha \sin \beta)\)

Use the angle addition formula for cosine

Recall that \(\cos(x - y) = \cos x \cos y + \sin x \sin y\). With this in mind, rewrite the expression from Step 3 as: \(2 + 2\cos(\alpha - \beta)\) Then, we can simplify this expression by factoring out 2: \(2\left(1 + \cos(\alpha - \beta)\right)\)

Use the double-angle formula for cosine

Recall that \(\cos(2x) = 2 \cos^2 x -1\). Solving for \(\cos^2 x\) gives us: \(\cos^2 x = \frac{1+\cos(2x)}{2}\) Apply this formula to the simplified expression in Step 4: \(2\left(\frac{1+ \cos(2\frac{\alpha - \beta}{2})}{2}\right)\) which simplifies to: \(4\cos^2\left(\frac{\alpha-\beta}{2}\right)\)

Compare the simplified expression with the given options

By comparing the final simplified expression with the given options, we find that it matches option (b). Thus, the correct answer to this exercise is: \((b) \:\: 4 \cos^2\left(\frac{\alpha-\beta}{2}\right)\)

Key concepts

Trigonometric Identities

In mathematics, trigonometric identities are equalities involving trigonometric functions that hold true for all the values of the variables within a certain domain. These identities are crucial in simplifying complex trigonometric expressions and solving trigonometric equations.

Some basic trigonometric identities include the reciprocal identities, like \( \sin(x) = \frac{1}{\csc(x)} \) and \( \cos(x) = \frac{1}{\sec(x)} \), and the quotient identities, such as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Mastering these relationships is key to progress in trigonometry, as they can transform complex equations into simpler forms that are easier to solve.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that for any angle \( x \), the square of the sine of \( x \) plus the square of the cosine of \( x \) equals one: \(\sin^2(x) + \cos^2(x) = 1\).

This identity is derived from the Pythagorean theorem, which relates the lengths of the sides of a right-angled triangle. It is particularly useful for converting between \( \sin \) and \( \cos \) terms or for proving other trigonometric identities. In the context of the original exercise, the Pythagorean identity was applied to simplify the terms \( \cos^2 \alpha \) and \( \sin^2 \alpha \) into a single numeric value, allowing for further simplification.

Angle Addition Formulas

Angle addition formulas play a vital role in trigonometry, allowing the simplification of expressions where the functions of a sum or difference of angles appear. The sine and cosine angle addition formulas are given by:\(
\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)
and
\(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\).

These formulas provide a method to break down complex trigonometric functions of a combined angle into simpler components. In the solution of the original exercise, the cosine angle addition formula was employed to rewrite \( 2\cos \alpha \cos \beta + 2\sin \alpha \sin \beta \) as \( 2\cos(\alpha - \beta) \) providing a crucial step in identifying the correct option.

Double-Angle Formulas

Double-angle formulas are specific cases of the angle addition formulas in which both angles are the same, effectively doubling the angle. The double-angle formulas include: \(\cos(2x) = \cos^2(x) - \sin^2(x)\), which can also be expressed as \(\cos(2x) = 2\cos^2(x) - 1\) or \(\cos(2x) = 1 - 2\sin^2(x)\). These relationships are indispensable when dealing with trigonometric expressions involving double angles.

In the context of our original exercise, the double-angle formula was used in the final steps to arrive at the answer. By recognizing that \( 1 + \cos(\alpha - \beta) \) can be represented as \( 2\cos^2(\frac{\alpha - \beta}{2}) \) with the proper manipulation, the solution could be matched with the correct answer as outlined.