Trigonometric equations

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Question

The number of real solutions of the equation $\sin (e^{x}) = 2^{x} + 2^{- x}$ is

Moderate

Solution

We know that
$\begin{array}{l} AM \geq GM \\ ∴ \frac{2^{x} + 2^{- x}}{2} \geq \sqrt{2^{x} \times 2^{- x}} \\ \Rightarrow 2^{x} + 2^{- x} \geq 2 \end{array}$
But, sin $(e^{x}) \leq 1$
Thus, $\sin (e^{x}) \neq 2^{x} + 2^{- x}$ for any $x \in R$
Hence, the given equation has no solution.

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