Show that the equation `e^(sinx)-e^(-sinx)-4=0` has no real solution.

Options

A. infinite number of real roots
B. no real roots
C. exactly one real root
D. exactly four real roots

Answer

2

Explanation

Let ` e^(sin x) =t`
` rArr t^(3) - 4t - 1 = 0 `
` t = (4 pmsqrt(16 + 4))/(2)`
`rArr t = e^(sin x ) = 2 pm sqrt(5)`
`rArr t = e^(sin x ) = 2 - sqrt(5),e^(sin x ) = 2 + sqrt(5)`
`rArr e^(sin x) = - sqrt(5) lt 0 ` , whihc is not possible
or ` sin x = In (2 + sqrt(5)) gt 1`, which is not possible
Hence no solution .