QJEEM12Ma62b5: JEE Main 2012 Mathematics - Class 11 JEEM12Ma62b5 Solution: The equation has:

Question

Verified Solution • 2025–26 Session moderate

The equation

{e^{\sin x}} - {e^{ - \sin x}} - 4 = 0

has:

Accepted Answer

Given equation is

{e^{\sin x}} - {e^{ - \sin x}} - 4 = 0

Put

{e^{{\mathop{\rm sinx}\nolimits} \,}} = t

in the given equation,

we get

{t^2} - 4t - 1 = 0

\Rightarrow t = {{4 \pm \sqrt {16 + 4} } \over 2}

\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm \sqrt {20} } \over 2}

\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm 2\sqrt 5 } \over 2}

\,\,\,\,\,\,\,\,\,\,\, = 2 \pm \sqrt 5

\Rightarrow {e^{\sin x}} = 2 \pm \sqrt 5

\,\,\,\,\,

(as

t = {e^{\sin x}}

)

\Rightarrow {e^{\sin x}} = 2 - \sqrt 5

and

\,\,\,\,\,\,\,\,\,\,\,\,\,

{e^{\sin x}} = 2 + \sqrt 5

\Rightarrow {e^{\sin x}} = 2 - \sqrt 5 &lt; 0

and

\,\,\,\,\,\,\sin x = \ln \left( {2 + \sqrt 5 } \right) &gt; 1

So, rejected

Hence given equation has no solution.

\therefore

The equation has no real roots.

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