Animated Solution for Mathematics - Functions: Show that the equation $e^{\sin x}-e^{-\sin x}-4=0$ has no real solution.

Welcome, future engineer. Today, we are going to dismantle a problem that, at first glance, looks like a chaotic mess of trigonometry and exponentials. You see an equation like $e^{\sin x}-e^{-\sin x}-4=0$ and your instinct might be to panic.

But I want you to take a deep breath. In the world of JEE Advanced, we don't panic; we observe. We look for the hidden symmetry.

The Algebraic Bridge

Look at the terms $e^{\sin x}$ and $e^{-\sin x}$. Do you see the relationship? One is the reciprocal of the other.

This is our golden ticket. Whenever you see a function and its reciprocal, your mind should immediately jump to the power of substitution. Let us define a new variable, $t=e^{\sin x}$.

By making this substitution, we are building a bridge from the world of transcendental functions—where things are curvy and unpredictable—to the world of algebra, where things are structured and solvable. Our equation transforms into:

Suddenly, the trigonometry has vanished, replaced by a clean, rational expression. To clear the fraction, we multiply the entire equation by $t$, giving us $t^2-1-4t=0$, or more standardly:

The Quadratic Crossroads

Now we are in familiar territory. We have a quadratic equation. Using the quadratic formula, $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, with $a=1,b=-4,c=-1$, we find the roots.

The discriminant is $(-4{)}^2-4(1)(-1)=16+4=20$. Thus, our roots are:

Here is where the trap lies. Many students stop here, thinking they have found the solution. But we must ask: are both these values physically possible for $t$?

Remember, $t=e^{\sin x}$. The exponential function $e^u$ is strictly positive for all real $u$. Since $\sqrt{5}\approx 2.236$, the root $2-\sqrt{5}$ is negative. We must reject it. The only candidate left is $t=2+\sqrt{5}\approx 4.236$.

The Reality Check

We have arrived at the final showdown. We need to solve $e^{\sin x}=2+\sqrt{5}$. Let's visualize this.

The left-hand side, $e^{\sin x}$, is a function of $x$. We know that for any real $x$, the sine function $\sin x$ is trapped between $-1$ and $1$. This means the exponent of our function is trapped between $-1$ and $1$.

Consequently, the function $e^{\sin x}$ is trapped between $e^{-1}$ and $e^1$. Calculating these bounds, we find that $e^{\sin x}$ oscillates strictly between approximately $0.368$ and $2.718$.

Now, look at our required value: $2+\sqrt{5}\approx 4.236$. The line $y=4.236$ is floating high above the maximum possible value of our function, which is $e\approx 2.718$.

The Elegance of Impossibility

There is no intersection. The curve of $e^{\sin x}$ never reaches the height of $4.236$.

It is a beautiful, elegant result. We have proven, through algebraic transformation and range analysis, that no real value of $x$ can satisfy this equation.

Mathematics is not just about finding an answer; it is about understanding the boundaries of what is possible. You have just mastered the art of bounding a function. Keep this intuition sharp, and no problem will ever be too intimidating again.