<p>The number of solutions of the equation \(1 + \sin^4 x = \cos^2(3x)\), \(x \in \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right]\) is</p>
Step-by-Step Solution
Key Concept: Use the ranges of both sides of the equation to determine when equality can hold. The equation is satisfied only when both sides equal their common value.
<p><strong>Solution:</strong> Given equation is $1 + \sin^4 x = \cos^2(3x)$</p><p>Since, range of $(1 + \sin^4 x) = [1, 2]$ and range of $\cos^2(3x) = [0, 1]$</p><p>So, the given equation holds if $1 + \sin^4 x = 1 = \cos^2(3x)$</p><p>$\Rightarrow \sin^4 x = 0$ and $\cos^2 3x = 1$</p><p>Since, $x \in \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right]$</p><p>$x = -2\pi, -\pi, 0, \pi, 2\pi$</p><p>Thus, there are five different values of $x$ possible.</p>
Correct Answer: B