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

Answer 1

$HEYA \: \\ \\ e {}^{ \sin(x) } - e {}^{ - \sin(x) } = 4 \\ \\ put \: \: e {}^{ \sin(x) } = z \: \: we \: have \: \\ \\ z {}^{2} - 4z - 1 = 0 \\ \\ z = 2 + \sqrt{5} \\ or \\ z = 2 - \sqrt{5} \\ \\ e {}^{ \sin(x) } = 2 + \sqrt{5} \\ \\ taking \: ln_{e} \: on \: both \: sides \: we \: have \: \\ \\ \sin( x) = ln(2 + \sqrt{5} ) \\ \\ \sin(x) = 1.46 \: \: \: \: (absurd \: ) \\ becoz \: range \: \: of \: sinx \: is \: ( - 1 \: to \: + 1) \\ \\ similarly \: \\ \sin(x) = ln(2 - \sqrt{5} ) \\ \\ \sin(x) = ln(- 0.23) \: \: (absurd \: ) \\ becoz \: domain \: of \\ \: ln \: is \: set \: of \: positive \: real \: numbers \: \\ \\ so \: given \: equation \: has \: no \: real \: roots \\ option \: b \: is \: correct$