If cos α = 1/2(a + 1/a), cos β = 1/2(b + 1/b), find the value of cos(α - β).

Question

If cos α = 1/2(a + 1/a), cos β = 1/2(b + 1/b), find the value of cos(α - β).

Asked on March 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-26T12:57:05.948Z

Step 1: Write down the given expressions for $\cos \alpha$ and $\cos \beta$. We are given: $$\cos \alpha = \frac{1}{2}\left(a + \frac{1}{a}\right)$$ $$\cos \beta = \frac{1}{2}\left(b + \frac{1}{b}\right)$$ Step 2: Find expressions for $\sin \alpha$ and $\sin \beta$. We use the identity $\sin^2 \theta = 1 - \cos^2 \theta$. First, calculate $\cos^2 \alpha$: $$\cos^2 \alpha = \left[\frac{1}{2}\left(a + \frac{1}{a}\right)\right]^2

Accepted Answer · 2026-03-26T12:57:05.948Z

Step 1: Write down the given expressions for $\cos \alpha$ and $\cos \beta$. We are given: $$\cos \alpha = \frac{1}{2}\left(a + \frac{1}{a}\right)$$ $$\cos \beta = \frac{1}{2}\left(b + \frac{1}{b}\right)$$ Step 2: Find expressions for $\sin \alpha$ and $\sin \beta$. We use the identity $\sin^2 \theta = 1 - \cos^2 \theta$. First, calculate $\cos^2 \alpha$: $$\cos^2 \alpha = \left[\frac{1}{2}\left(a + \frac{1}{a}\right)\right]^2