Question

The equation e^sinx - e^(-sinx)- 4 = 0 has
(a) infinite number of real roots
(b) no real roots
(c) exactly one real root
(d) exactly four real roots
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Answer 1

$HEYA \: \\ \\ e {}^{ \sin(x) } - 1 \div e {}^{ \sin(x) } = 4 \\ \\ put \: \: e {}^{ \sin(x) } = z \: \: we \: \: have \: \\ \\ z {}^{2} - 1 = 4z \\ \\ z {}^{2} - 4z - 1 = 0 \\ \\ z = 2 + \sqrt{5} \: \: or \: \: z = 2 - \sqrt{5} \\ \\ e {}^{ \sin(x) } = 2 + \sqrt{5} \: \: \: or \: \: \: e {}^{ \sin(x) } = 2 - \sqrt{5} \\ \\ taking \: log_{e} \: on \: both \: side \: we \: have \: \\ \\ \sin(x) = log_{e}(2 + \sqrt{5} ) \: or \: \sin(x) = 2 - \sqrt{5} \\ \\ \sin(x) = 1.44 \:(absurd \: ) \: or \: \: \sin(x) = log_{e}( - 0.23) \: (absurd \: ) \\ \\ so \: given \: equation \: has \: no \: real \: roots. \\ option \: b \: is \: correct$