First slide

Theory of expressions

Question

The number of real roots of the equation $e^{\sin x} - e^{- \sin x} - 4 = 0$ are

Moderate

A

1
B

2
C

Infinite
D

None of these

Solution

. Given equation is $e^{\sin x} - e^{- \sin x} - 4 = 0$
Let $e^{\sin x} = y$ ,then given equation can be written as
$\begin{aligned} y^{2} - 4 y - 1 = 0 \\ \Rightarrow y = 2 \pm \sqrt{5} \end{aligned}$
But the value of $y = e^{\sin x}$ ' is always positive, so we take only
$\begin{aligned} y = 2 + \sqrt{5} \\ \Rightarrow \log_{e} y = \log_{e} (2 + \sqrt{5}) \\ \Rightarrow \sin x = \log_{e} (2 + \sqrt{5}) > 1 \end{aligned}$
Which is impossible since sin x cannot be greater
than 1.
Hence, we cannot find any real value of x which satisfies
the given equation.

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