Question

The period of \(f(x) = \sin\left(\frac{\pi x}{n-1}\right) + \cos\left(\frac{\pi x}{n}\right)\), \(n \in \mathbb{Z}\), \(n>2\) is

• A. \(2\pi n(n-1)\)
• B. \(4n(n-1)\)
• C. \(2n(n-1)\)
• D. None of the above

Hint:

Period of \(\sin(ax)\) = \(\frac{2\pi}{|a|}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Period = LCM of individual periods.
Step 2: Detailed Explanation:
Period of \(\sin(\pi x/(n-1))\): \(T_1 = 2(n-1)\)
Period of \(\cos(\pi x/n)\): \(T_2 = 2n\)
LCM of \(2(n-1)\) and \(2n = 2n(n-1)\)
Step 3: Final Answer:
\(2n(n-1)\).