Question 263 of 298MediumMCQ
The equation esinx−e−sinx−4=0 has:
Solutions · 1
Official solution
Given equation is \[{e^{\sin x}} - {e^{ - \sin x}} - 4 = 0\]
Put \[{e^{{\mathop{\rm sinx}\nolimits} \,}} = t\] in the given equation,
we get t2−4t−1=0
\[ \Rightarrow t = {{4 \pm \sqrt {16 + 4} } \over 2}\]
\[\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm \sqrt {20} } \over 2}\]
\[\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm 2\sqrt 5 } \over 2}\]
\[\,\,\,\,\,\,\,\,\,\,\, = 2 \pm \sqrt 5 \]
\[ \Rightarrow {e^{\sin x}} = 2 \pm \sqrt 5 \] \[\,\,\,\,\,\] (as \[t = {e^{\sin x}}\])
\[ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 \] and
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\] \[{e^{\sin x}} = 2 + \sqrt 5 \]
\[ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 < 0\]
and \[\,\,\,\,\,\,\sin x = \ln \left( {2 + \sqrt 5 } \right) > 1\] So, rejected
Hence given equation has no solution.
\[\therefore\] The equation has no real roots.
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