Question

The period of the function \( f(x) = \cos 4x + \tan 3x \) is

• A. \( \frac{\pi}{12} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{2} \)
• D. \( \pi \)
• E. \( 2\pi \)

Hint:

For sum of periodic functions, always take LCM of individual periods.

Answer 1

The Correct Option is D

Concept: \[ \text{Period of } \cos(ax) = \frac{2\pi}{a}, \quad \text{Period of } \tan(ax) = \frac{\pi}{a} \]

Step 1: Period of each function
\[ \cos 4x \Rightarrow \frac{2\pi}{4} = \frac{\pi}{2} \] \[ \tan 3x \Rightarrow \frac{\pi}{3} \]

Step 2: Find LCM of periods
We need smallest \(T\) such that: \[ T = n \cdot \frac{\pi}{2} = m \cdot \frac{\pi}{3} \]

Step 3: Solve
LCM of \( \frac{\pi}{2} \) and \( \frac{\pi}{3} \) is: \[ \pi \]

Step 4: Verification
\[ \cos 4(x+\pi) = \cos(4x+4\pi) = \cos 4x \] \[ \tan 3(x+\pi) = \tan(3x+3\pi) = \tan 3x \]

Step 5: Final Answer
\[ \boxed{\pi} \]