First slide

Theory of expressions

Question

If $e^{\cos x} - e^{- \cos x} = 4$ then the value of $\cos x$ , is

Moderate

A

$\log_{e} (2 + \sqrt{5})$
B

$- \log_{e} (2 + \sqrt{5})$
C

$\log_{e} (- 2 + \sqrt{5})$
D

none of these

Solution

Let $e^{\cos x} = y$ . Then ,
$\begin{array}{l} e^{\cos x} - e^{- \cos x} = 4 \\ \Rightarrow y - \frac{1}{y} = 4 \\ \Rightarrow y^{2} - 4 y - 1 = 0 \\ \Rightarrow y = 2 \pm \sqrt{5} \\ \Rightarrow y = 2 + \sqrt{5} as y > 0 \\ \Rightarrow e^{\cos x} = 2 + \sqrt{5} \Rightarrow \cos x = \log_{e} (2 + \sqrt{5}) \end{array}$
Clearly , $\log_{c} (2 + \sqrt{5}) > 1$ and $\cos x \leq 1$
So, there is no value of $\cos x$ satisfying the given equation

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