Animated Solution for Mathematics - Functions: The number of solutions of the equation $\sin (e^x)=5^x+5^{-x}$ is

A Mathematical Standoff

Imagine you are standing on a vast, flat plain. On one side, you have a function that is constantly oscillating, a rhythmic wave that refuses to rise above a certain height. On the other side, you have a function that is growing, stretching its arms toward infinity, never dipping below a specific floor.

The question is: do they ever meet? In the world of JEE Advanced, we often encounter equations that look intimidating—like $\sin (e^x)=5^x+5^{-x}$—but the secret to conquering them isn't brute force; it is observation.

Let us look at the left side of our equation: $f(x)=\sin (e^x)$. Many students see the $e^x$ and immediately panic, thinking about derivatives or complex transcendental behavior. But take a deep breath.

What is the soul of the sine function? No matter what you feed into it—whether it is $x$, $x^2$, or even $e^x$—the sine function is a prisoner of its own geometry. It is defined by the unit circle, and its output is strictly bounded: $-1\le \sin (\theta )\le 1$.

Because $e^x$ is always a real number for any real $x$, the expression $\sin (e^x)$ is trapped forever in the interval $[-1,1]$. It can never, ever exceed $1$.

Now, turn your gaze to the right side: $g(x)=5^x+5^{-x}$. This is a classic structure. We have a term and its reciprocal.

Whenever you see $a+\frac{1}{a}$ where $a>0$, your mind should immediately jump to the Arithmetic Mean-Geometric Mean (AM-GM) inequality. The inequality states that for any positive numbers $a$ and $b$, $\frac{a+b}{2}\ge \sqrt{ab}$.

If we set $a=5^x$ and $b=5^{-x}$, we get:

Since $5^x\cdot 5^{-x}=5^0=1$, the inequality simplifies beautifully to:

This tells us that the RHS has a 'floor'. It can be $2$, it can be $100$, it can be a billion, but it can never be less than $2$. The RHS lives in the interval $[2,\infty )$.

The Non-Overlap

Now, we bring them together. We have the LHS, which is desperately trying to reach higher than $1$, and the RHS, which refuses to descend below $2$.

There is a clear, undeniable gap between these two worlds. The maximum value the LHS can ever achieve is $1$, while the minimum value the RHS can ever achieve is $2$.

Because $1<2$, there is no real value of $x$ that can satisfy this equality.

This problem is a masterclass in Range Analysis. In the heat of an exam, you might be tempted to try and solve for $x$ using logarithms or complex identities, but that is a path to frustration.

The beauty of mathematics lies in knowing when to stop calculating and start observing. By identifying the bounds of each function, we have solved the problem without ever needing to find a specific value for $x$.

We have proven that the set of solutions is empty. The number of solutions is $0$. Remember, sometimes the most powerful answer is realizing that the event you are looking for is, quite simply, impossible.