The number of solutions of the equation is trigonometry equation

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Solution

Q.
The number of solutions of the equation $e^{\sin x} - 2 e^{- \sin x} = 2,$ is [2024]

1 0

2 2

3 1

4 more than 2

Ans.
(1)

We have, $e^{\sin x} - 2 e^{- \sin x} = 2$ ...(i)

Let $e^{\sin x} = t$

$∴$ $(i) becomes$ $t - \frac{2}{t} = 2$

$\Rightarrow t^{2} - 2 = 2 t \Rightarrow t^{2} - 2 t - 2 = 0$

$\Rightarrow t = \frac{2 \pm \sqrt{4 - 4 (1) (- 2)}}{2} \Rightarrow t = \frac{2 \pm \sqrt{12}}{2} = 1 \pm \sqrt{3}$

Now, $e^{\sin x} = 1 + \sqrt{3} and e^{\sin x} = 1 - \sqrt{3} (Not possible)$

$\Rightarrow \sin x = \ln (1 + \sqrt{3}) > 1 (Not possible)$

$∴$ $There is no solution.$

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