Question

e^(sinx)-e^(-sinx)-4=0

Then equation has solutions:

a) 1

b) Many solutions

c) 2

d) No solution

Solution

Maximum value of e^sinx will be e= 2.71, because -1<=sinx<=1. So, maximum value of e^sinx- e^-sinx is less than 4. Therefore no solution exist.

Alternative

Put e^(sinx)= t

=> t^2-4t-1=0

=> t = (4+-sqrt(20))/2

=> t= 2+-sqrt5

t= 2-sqrt5 is not possible,because e^(sinx) can't be negative.

t= 2+sqrt5=4.23 also can't be possible. Maximum value of sinx is 1. Therefore maximum value of e^(sinx) will be e=2.71

So, there is no solution exist for given equation.