Question

The values of \( x \) in \( (-\pi, \pi) \) which satisfy the equation \( \cos x + \cos 2x + \cos 3x + \cdots = 4^3 \) are:

• A. \( \pm \frac{\pi}{4}, \pm \frac{3\pi}{4} \)
• B. \( \pm \frac{\pi}{6}, \pm \frac{\pi}{3} \)
• C. \( \pm \frac{\pi}{8} \)
• D. \( \frac{\pi}{3} \)

Hint:

For solving trigonometric series or equations involving multiple cosines, recognize patterns or use known trigonometric identities to simplify and solve.

Answer 1

The Correct Option is A

The given equation is: \[ \cos x + \cos 2x + \cos 3x + \cdots = 4^3. \] This is an infinite sum of cosines, which can be represented using a standard identity for the sum of cosines in an infinite series. By simplifying this equation and solving for the values of \( x \) that satisfy it, we find that the possible solutions for \( x \) are: \[ x = \pm \frac{\pi}{4}, \pm \frac{3\pi}{4}. \]