The equation e sin x - e - sin x - 4 = 0 has

Question

The equation $e^{\sin x} - e^{- \sin x} - 4 = 0$ has

Smart Achievers · Accepted Answer

(2)

Given equation is

$e^{\sin x} - e^{- \sin x} = 4$ $\Rightarrow e^{\sin x} - \frac{1}{e^{\sin x}} = 4$

Now, let $y = e^{\sin x}$

Then, we get

$y - \frac{1}{y} = 4$ $\Rightarrow y^{2} - 4 y - 1 = 0$

$∴$ $y = \frac{4 \pm \sqrt{16 + 4}}{2} = 2 \pm \sqrt{5}$

$\Rightarrow e^{\sin x} = 2 \pm \sqrt{5}$

Since sine is a bounded function, i.e., $- 1 \leq \sin x \leq 1 .$

Therefore, we get $e^{- 1} \leq e^{\sin x} \leq e$

$\Rightarrow e^{\sin x} \in [\frac{1}{e}, e]$

Also, it is obvious that $2 + \sqrt{5} > e$

and $2 - \sqrt{5} < \frac{1}{e}$ $\Rightarrow 2 \pm \sqrt{5} \notin [\frac{1}{e}, e]$

So, $e^{\sin x} = 2 + \sqrt{5}$ is not possible for any $x \in R$ ,

and $e^{\sin x} = 2 - \sqrt{5}$ is also not possible for any $x \in R$

Hence, we can say that the given equation has no solution.

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