Finding cos 2β using Geometric Mean

This problem involves trigonometric concepts and the definition of the Geometric Mean (GM).

We are given two key pieces of information:

• sin β is the Geometric Mean (GM) of sin α and cos α.
• tan γ is the Arithmetic Mean (AM) of sin α and cos α. (Note: This piece of information about γ is not needed to solve for cos 2β).

Understanding Geometric Mean

For two positive numbers, 'a' and 'b', their Geometric Mean (GM) is defined as \(\sqrt{ab}\). If we are told that a third number 'g' is the GM of 'a' and 'b', it means \(g = \sqrt{ab}\), which implies \(g^2 = ab\).

Applying GM to the Problem

According to the problem statement, sin β is the GM of sin α and cos α. Therefore, we can write the relationship as:

\((\text{sin } \beta)^2 = (\text{sin } \alpha)(\text{cos } \alpha)\)

So, \(\sin^2 \beta = \sin \alpha \cos \alpha\).

Finding cos 2β

We need to find the value of cos 2β. We can use the double angle identity for cosine that involves \(\sin^2 \beta\). The identity is:

\(\cos 2\beta = 1 - 2 \sin^2 \beta\)

Substituting the Value of sin² β

We found that \(\sin^2 \beta = \sin \alpha \cos \alpha\) from the GM definition. Let's substitute this into the formula for cos 2β:

\(\cos 2\beta = 1 - 2 (\sin \alpha \cos \alpha)\)

\(\cos 2\beta = 1 - 2 \sin \alpha \cos \alpha\)

Comparing with Options

Now we need to see which of the given options matches the expression \(1 - 2 \sin \alpha \cos \alpha\). Let's look at the first option:

Option 1: \((\cos \alpha-\sin \alpha)^2\)

Let's expand this expression:

\((\cos \alpha-\sin \alpha)^2 = \cos^2 \alpha - 2 (\cos \alpha)(\sin \alpha) + \sin^2 \alpha\)

Rearranging and using the identity \(\cos^2 \alpha + \sin^2 \alpha = 1\):

\((\cos \alpha-\sin \alpha)^2 = (\cos^2 \alpha + \sin^2 \alpha) - 2 \sin \alpha \cos \alpha\)

\((\cos \alpha-\sin \alpha)^2 = 1 - 2 \sin \alpha \cos \alpha\)

This matches the expression we found for cos 2β.

Let's quickly check the other options for completeness:

Option 2: \((\cos \alpha+\sin \alpha)^2 = \cos^2 \alpha + 2 \sin \alpha \cos \alpha + \sin^2 \alpha = 1 + 2 \sin \alpha \cos \alpha\). This is not equal to \(1 - 2 \sin \alpha \cos \alpha\).

Option 3: \((\cos \alpha-\sin \alpha)^3\). Expanding this would result in a polynomial in terms of sin α and cos α with degree 3, not a simple linear combination of 1 and \( \sin \alpha \cos \alpha \).

Option 4: \(\frac{(\cos \alpha-\sin \alpha)^2}{2} = \frac{1 - 2 \sin \alpha \cos \alpha}{2}\). This is not equal to \(1 - 2 \sin \alpha \cos \alpha\).

Therefore, the correct option is \((\cos \alpha-\sin \alpha)^2\).

Step-by-Step Derivation

1. Start with the definition of sin β being the GM of sin α and cos α: \(\sin^2 \beta = \sin \alpha \cos \alpha\).
2. Recall the double angle identity for cosine: \(\cos 2\beta = 1 - 2 \sin^2 \beta\).
3. Substitute the expression for \(\sin^2 \beta\) into the identity: \(\cos 2\beta = 1 - 2 (\sin \alpha \cos \alpha)\).
4. Consider the first option: \((\cos \alpha-\sin \alpha)^2\).
5. Expand the option: \((\cos \alpha-\sin \alpha)^2 = \cos^2 \alpha - 2\sin\alpha\cos\alpha + \sin^2 \alpha\).
6. Use the identity \(\cos^2 \alpha + \sin^2 \alpha = 1\): \((\cos \alpha-\sin \alpha)^2 = 1 - 2\sin\alpha\cos\alpha\).
7. Compare the result from step 3 and step 6: \(\cos 2\beta = 1 - 2 \sin \alpha \cos \alpha\) and \((\cos \alpha-\sin \alpha)^2 = 1 - 2 \sin \alpha \cos \alpha\).
8. They are equal, so \(\cos 2\beta = (\cos \alpha-\sin \alpha)^2\).

Revision Table: Key Trigonometric Concepts

Additional Information: Means and Trigonometry

This question connects the concept of means (Arithmetic Mean and Geometric Mean) with trigonometry. Understanding different types of means is important in mathematics.

• Arithmetic Mean (AM): For two numbers a and b, \(\text{AM} = \frac{a+b}{2}\). This is the average.
• Geometric Mean (GM): For two positive numbers a and b, \(\text{GM} = \sqrt{ab}\). This is used in geometric sequences and calculating average growth rates.
• Harmonic Mean (HM): For two numbers a and b, \(\text{HM} = \frac{2}{\frac{1}{a}+\frac{1}{b}} = \frac{2ab}{a+b}\). This is used in rates and ratios.

There is a relationship between these means: For positive numbers, \(AM \ge GM \ge HM\).

Trigonometric identities are fundamental for simplifying expressions and solving equations in trigonometry. Double angle formulas, like \(\cos 2\beta = 1 - 2 \sin^2 \beta\), are derived from the sum identities (\(\cos(A+B)\)) and are very useful in many problems.

In this specific problem, recognizing the identity \(1 - 2 \sin \alpha \cos \alpha = 1 - \sin 2\alpha\) is also useful, although the options were in terms of \(\cos \alpha\) and \(\sin \alpha\) squared. Also, recognizing the expansion of \((\cos \alpha - \sin \alpha)^2\) was key to matching the options.