Animated Solution for Mathematics - Trigonometry: If $f(x)=\cos (\ln x)$, then $f(x)f(y)-\frac{1}{2}[f(x/y)+f(xy)]$ has the value

Welcome, future engineer! Today, we are going to unravel a problem that might look like a daunting, tangled mess of functions, but is actually a masterclass in mathematical elegance.

We are given the function $f(x)=\cos (\ln x)$ and asked to evaluate the expression:

At first glance, this expression seems intimidating. A cosine of a logarithm feels like we are mixing two different worlds, but in the world of JEE Advanced, these combinations are often designed to reveal beautiful, hidden symmetries.

Deconstructing the Function

Our first step is to understand the components of our expression. We have $f(x)=\cos (\ln x)$.

To evaluate the terms $f(x/y)$ and $f(xy)$, we substitute the arguments into our function:

We now apply the fundamental properties of logarithms: $\ln (x/y)=\ln x-\ln y$ and $\ln (xy)=\ln x+\ln y$.

Substituting these into our terms, we obtain:

The Trigonometric Bridge

Now, let us look at the full expression $E$ again:

Look closely at the terms inside the square bracket. We have $\cos (A-B)+\cos (A+B)$, where $A=\ln x$ and $B=\ln y$.

This is a classic trigonometric pattern. We recall the standard identity:

This identity is the bridge that connects our logarithmic expansion to the final simplification.

The Grand Cancellation

With the identity in hand, we simplify the bracketed term. Substituting $A=\ln x$ and $B=\ln y$, the expression inside the bracket becomes:

Now, let us substitute this back into our main expression $E$:

Notice that the $\frac{1}{2}$ and the $2$ cancel each other out perfectly. We are left with:

The entire expression collapses to 0. It is incredibly satisfying to see such a complex-looking problem simplify to such a neat, round number.