Question

If \( \cos\theta=\frac{1}{2}\left(a+\frac{1}{a}\right) \), then \(4\cos^3\theta-3\cos\theta=\)

• A. \(a^3+\frac{1}{a^3}\)
• B. \(\frac{1}{2}\left(a^3+\frac{1}{a^3}\right)\)
• C. \(\frac{1}{4}\left(a^3+\frac{1}{a^3}\right)\)
• D. \(\frac{1}{3}\left(a^3+\frac{1}{a^3}\right)\)

Hint:

Whenever you see \(4\cos^3\theta-3\cos\theta\), immediately remember it is equal to \(\cos 3\theta\).

Answer 1

The Correct Option is B

Concept: We use the triple angle identity: \[ \cos 3\theta=4\cos^3\theta-3\cos\theta \]

Step 1: Given: \[ \cos\theta=\frac{1}{2}\left(a+\frac{1}{a}\right) \]

Step 2: We need to find: \[ 4\cos^3\theta-3\cos\theta \] Using the identity: \[ 4\cos^3\theta-3\cos\theta=\cos 3\theta \]

Step 3: If \[ \cos\theta=\frac{1}{2}\left(a+\frac{1}{a}\right) \] then by the standard expansion form: \[ \cos 3\theta=\frac{1}{2}\left(a^3+\frac{1}{a^3}\right) \]

Step 4: Hence: \[ 4\cos^3\theta-3\cos\theta = \frac{1}{2}\left(a^3+\frac{1}{a^3}\right) \] Therefore, \[ \boxed{\frac{1}{2}\left(a^3+\frac{1}{a^3}\right)} \]