Question

If \( 0 \le x \le 2\pi \), then the number of solutions of the equation \( \sin^8 x + \cos^8 x = 1 \) is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(5\)
• E. \(8\)

Hint:

High even powers often reduce to checking extreme values (0 or 1).

Answer 1

The Correct Option is C

Concept: \[ \sin^2 x + \cos^2 x = 1 \]

Step 1: Use identity
\[ \sin^8 x + \cos^8 x = (\sin^4 x)^2 + (\cos^4 x)^2 \]

Step 2: Use inequality
Maximum occurs when one term is 1 and other is 0.

Step 3: Solve condition
\[ \sin^8 x + \cos^8 x = 1 \Rightarrow \sin x = 0 \text{ or } \cos x = 0 \]

Step 4: Solutions
\[ \sin x = 0 \Rightarrow x = 0,\pi,2\pi \] \[ \cos x = 0 \Rightarrow x = \frac{\pi}{2},\frac{3\pi}{2} \]

Step 5: Count distinct solutions
\[ 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \] \[ \Rightarrow 4 \text{ solutions} \]

Step 6: Final Answer
\[ \boxed{4} \]