So many roots?

e^sinx can have a maximum value of e^1=2.7 but in problem 4 and e^-sinx have been subtracted. which will give a negative value always since e^-sinx will be always positive. hence whole expression will be always negative and never be equal to zero..........

Let ${ e }^{ \sin { x } }=y$

then, $y-\frac { 1 }{ y } -4=0$ which on solving gives us $y=2\pm \sqrt { 5 }$

so, ${ e }^{ \sin { x } }=2+\sqrt { 5 }$ ..........( as ${ e }^{ \sin { x } }$ cannot be $<0$ )

$\Rightarrow \quad \sin { x\log _{ e }{ e } } =\quad \log _{ e }{ \left( 2+\sqrt { 5 } \right) }$ .....................(i)

now, $2+\sqrt { 5 } > e$

$\Rightarrow \log _{ e }{ \left( 2+\sqrt { 5 } \right) } >\log _{ e }{ e }$

$\Rightarrow \sin { x>1 }$ which is not possible as $\sin { x\in } \left[ -1,1 \right]$

so, the equation has no real solutions

let y = e^sinx , then y^2 -4y-1 = 0 , so , by complete the square ( or no need do this step ) , we know that this function nt touch the x-axis at all ... so , y = nt real root , then x must nt hv real root also .