$e^{s i n x} - e^{- s i n x} = 4$ Find the solution?

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Solution

$e^{s i n x} - e^{- s i n x} = 4$
Let $e^{s i n x} = y$
$∴ y - \frac{1}{y} = 4 \Rightarrow y^{2} - 1 = 4 y \Rightarrow y^{2} - 4 y - 1 = 0 D = 16 - 4 (- 1) = 20 y = \frac{4 \pm \sqrt{20}}{2} y = \frac{4 + \sqrt{20}}{2} = 4.236 a n d \frac{4 - \sqrt{20}}{2} = - 0.236$
we know that $y = e^{s i n x}$
which can never be -ve.
Aslo, $(s i n x)_{m a x} = 1$
$∴ y_{m a x} = e^{2} = 3 = 2.718$
So, neither of the values of y are possible
so, no solution

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