<p>The number of solutions of the equation \(e^{\sin x} - e^{-\sin x} = 4\) is</p>
Step-by-Step Solution
Key Concept: Recognize that f(t) = e^t - e^(-t) is strictly increasing, so for each value of sin x, there's at most one solution. The equation requires e^(sin x) - e^(-sin x) = 4, which means we need to find what value of sin x satisfies this, then check if that value is achievable since -1 ≤ sin x ≤ 1.
<p><strong>Step 1:</strong> Let f(t) = e^t - e^(-t). This function is strictly increasing on ℝ since f'(t) = e^t + e^(-t) > 0 for all t.</p><p><strong>Step 2:</strong> For the equation e^(sin x) - e^(-sin x) = 4 to have solutions, we need sin x to satisfy e^(sin x) - e^(-sin x) = 4.</p><p><strong>Step 3:</strong> Find the maximum possible value. Since -1 ≤ sin x ≤ 1, the maximum of e^(sin x) - e^(-sin x) occurs at sin x = 1:<br/>e^1 - e^(-1) = e - 1/e ≈ 2.718 - 0.368 ≈ 2.35</p><p><strong>Step 4:</strong> Since 2.35 < 4, the equation e^(sin x) - e^(-sin x) = 4 has no solutions in the valid domain where sin x ∈ [-1, 1].</p><p>∴ Answer: <strong>B (0 solutions)</strong></p>
Correct Answer: B