Animated Solution for Mathematics - Functions: If $f(x)=\cos [{\pi }^2]x+\cos [-{\pi }^2]x$, where $[x]$ stands for the greatest integer function, then

Welcome, fellow traveler, to the fascinating world of the Greatest Integer Function (GIF). Often, students see these square brackets and feel a sudden, cold shiver of intimidation.

But today, we are going to strip away that fear. We are going to look at the function $f(x)=\cos ([{\pi }^2]x)+\cos ([-{\pi }^2]x)$ and realize that it is not a monster, but a beautifully structured puzzle waiting to be solved.

Demystifying the Constants

Before we can even touch the trigonometry, we must confront the coefficients. The GIF, denoted by $[x]$, is the floor function, which is the largest integer less than or equal to $x$.

Think of it as a 'left-pointing' arrow on the number line. If you are standing at $9.8696$, you look to your left, and the first integer you hit is $9$. So, $[{\pi }^2]=[9.8696]=9$.

Now, here is where the trap lies: what about $[-{\pi }^2]$? We are looking at $-9.8696$.

If you are standing at $-9.8696$ on the number line, looking to the left means moving toward $-10,-11,-12$. The first integer to your left is $-10$.

This is the most common mistake in JEE exams: assuming the GIF of a negative number is just the negative of the positive GIF. It is not; it is $-10$.

The Elegant Simplification

With our constants secured, the function transforms. We replace the brackets with our hard-earned integers:

Now, we invoke the power of trigonometry. We know that cosine is an even function, meaning it is symmetric about the y-axis. Mathematically, this gives us the beautiful property $\cos (-\theta )=\cos (\theta )$.

Thus, $\cos (-10x)$ simply becomes $\cos (10x)$. Our function is now a clean, manageable expression:

The Final Verification

Now, we test our options. Let us look at $f(\frac{\pi }{2})$. Substituting this into our function, we get:

This simplifies to $\cos (\frac{9\pi }{2})+\cos (5\pi )$. Since $\frac{9\pi }{2}$ is an odd multiple of $\frac{\pi }{2}$, its cosine is $0$. Since $5\pi$ is an odd multiple of $\pi$, its cosine is $-1$.

Thus, $f(\frac{\pi }{2})=0-1=-1$. Option A is correct!

Next, we check $f(-\pi )=\cos (-9\pi )+\cos (-10\pi )$. Again, using the even property, this is $\cos (9\pi )+\cos (10\pi )$.

Since $9\pi$ is an odd multiple of $\pi$, its cosine is $-1$. Since $10\pi$ is an even multiple of $\pi$, its cosine is $1$.

Thus, $f(-\pi )=-1+1=0$. Option C is also correct!

We have navigated the traps, simplified the expression, and verified our results. The lesson here is simple: never let the notation intimidate you.

Whether it is the Greatest Integer Function or a complex trigonometric identity, break it down into its core components, visualize the number line, and trust your process. You have the tools; now go forth and conquer.