Question

Given below are pair of functions \(f(x)\) and \(g(x)\). Which pair is not linearly independent?

• A. \(f(x) = \cos x + \sin x\), \(g(x) = -6\cos\frac{x}{3} - 8\cos^3\frac{x}{3}\)
• B. \(f(x) = -\cos x + \sin x\), \(g(x) = \cos x\)
• C. \(f(x) = \cos x\), \(g(x) = \cos^3 x - \sin^3 x\)
• D. \(f(x) = \sin x\), \(g(x) = 6\sin\frac{x}{3} - 8\sin^3\frac{x}{3}\)

Hint:

For function pairs, look for standard trigonometric expansions like \(\sin(3\theta)\) or \(\cos(3\theta)\).
If \(g(x) = c \cdot f(x)\) holds for all \(x\), the Wronskian is zero, meaning they are not linearly independent.

Answer 1

The Correct Option is D

Concept:
• Two functions \(f(x)\) and \(g(x)\) are linearly dependent (not linearly independent) if one is a constant multiple of the other, i.e., \(g(x) = c \cdot f(x)\).
• Use the triple-angle trigonometric identity: \(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\).

Step 1: Analyze option D using triple-angle identity
Let \(\theta = \frac{x}{3}\). Then \(3\theta = x\).
The identity gives: \[ \sin x = 3\sin\frac{x}{3} - 4\sin^3\frac{x}{3} \]

Step 2: Relate \(g(x)\) to \(f(x)\) for option D
The given function is: \[ g(x) = 6\sin\frac{x}{3} - 8\sin^3\frac{x}{3} \] Factor out a 2: \[ g(x) = 2\left(3\sin\frac{x}{3} - 4\sin^3\frac{x}{3}\right) \] Substitute the identity from
Step 1: \[ g(x) = 2\sin x = 2f(x) \] Since \(g(x)\) is a direct linear multiple of \(f(x)\), they are linearly dependent.