The equation `e^(sin x) -e^(-sin x)-4=0` has

Options

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

Answer

B

Explanation

Let ` e^(sin x) = t`
` rArr t^(2) - 4t - 1= 0`
` rArr t = (4 pm sqrt(16+4))/2`
` rArr t=e^(sin x) = 2 pm sqrt5`
` rArr e^(sin x) = 2 - sqrt5, e^(sin x ) = 2 + sqrt 5 `
` e ^(sin x ) = 2 - sqrt 5 lt 0 ` ,
` rArr sin x = "In " (2+sqrt5) gt 1`
So it is rejected, hence there is no solution.