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The given equation is:

\[
e^{\sin x} - e^{-\sin x} - 4 = 0
\]

We can simplify this using the identity for hyperbolic functions:

\[
e^{\sin x} - e^{-\sin x} = 2 \sinh(\sin x)
\]

Thus, the equation becomes:

\[
2 \sinh(\sin x) - 4 = 0
\]

Solving for \(\sinh(\sin x)\):

\[
2 \sinh(\sin x) = 4
\]

\[
\sinh(\sin x) = 2
\]

Next, we solve for \(\sin x\) by finding the inverse hyperbolic sine (arsinh):

\[
\sin x = \sinh^{-1}(2)
\]

The value of \(\sinh^{-1}(2)\) is approximately 1.443. Therefore:

\[
\sin x = 1.443
\]

However, \(\sin x\) can only take values between -1 and 1. Since \(\sin x = 1.443\) exceeds this range, there is **no real solution** for \(x\).

Hence, the correct answer is:

**(d) None of these**

Would you like further explanation or have any questions?

Here are five related questions:
1. What are the properties of the hyperbolic sine function \(\sinh(x)\)?
2. Can you explain how inverse hyperbolic functions are calculated?
3. What is the range of the sine function, and why is it important in this context?
4. How can we check if a transcendental equation like this has real roots without solving it fully?
5. What is the difference between hyperbolic functions and trigonometric functions?

**Tip:** When dealing with transcendental equations, it's useful to check the ranges of the functions involved to quickly determine if real solutions are possible.

The number of real roots of the equation e^(sin x) - e^(-sin x) - 4 = 0 are?

This problem involves solving a transcendental equation using hyperbolic identities. The key steps include simplifying the equation with sinh(x) and verifying the solution using the range of the sine function. Suitable for undergraduate calculus students.

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