Animated Solution for Mathematics - Trigonometry: The value of ${\cos }^3\frac{\pi }{8}\cos \frac{3\pi }{8}+{\sin }^3\frac{\pi }{8}\sin \frac{3\pi }{8}$ is:

Welcome, my dear student. Today, we are going to peel back the layers of a trigonometric expression that, at first glance, looks like a chaotic mess of powers and angles.

You see ${\cos }^3\frac{\pi }{8}\cos \frac{3\pi }{8}+{\sin }^3\frac{\pi }{8}\sin \frac{3\pi }{8}$ and your instinct might be to reach for a calculator or panic. But I want you to take a deep breath. In the world of JEE Advanced, we don't calculate; we observe. We look for the hidden architecture beneath the numbers.

The Power of Substitution

The first thing that should strike you is the repetition of $\frac{\pi }{8}$. It is the heartbeat of this problem.

Let us define $\theta =\frac{\pi }{8}$. Suddenly, the expression becomes:

Do you see what happened? We have stripped away the numerical clutter and revealed the underlying structure. We are no longer dealing with specific angles; we are dealing with a general relationship between $\theta$ and $3\theta$. This is the first step in mastering advanced mathematics: abstraction.

The Triple Angle Arsenal

Now, we face the terms $\cos 3\theta$ and $\sin 3\theta$. These are the 'monsters' of the expression. But every monster has a weakness.

For trigonometry, that weakness is the triple angle identity. We know that:

By substituting these into our expression, we are essentially breaking the complex angles down into their fundamental components. The expression transforms into:

The Algebraic Dance

Now comes the part where many students lose their way. We must expand these terms carefully.

Distributing ${\cos }^3\theta$ gives us $4{\cos }^6\theta -3{\cos }^4\theta$. Distributing ${\sin }^3\theta$ gives us $3{\sin }^4\theta -4{\sin }^6\theta$.

When we combine them, we get:

This is where the elegance begins. We have grouped the terms by their coefficients. We are looking at a difference of powers.

We know that ${\cos }^4\theta -{\sin }^4\theta$ is a difference of squares, which simplifies to $({\cos }^2\theta -{\sin }^2\theta )({\cos }^2\theta +{\sin }^2\theta )$. Since ${\cos }^2\theta +{\sin }^2\theta =1$, this collapses into $\cos 2\theta$.

The power of 6 term is slightly more complex, but it follows the same logic of factoring. It is a dance of identities, and you are leading the steps.

The Final Collapse

After the dust settles from the algebra, we find ourselves with $\cos 2\theta (1-4{\cos }^2\theta {\sin }^2\theta )$. Look closely at that term $4{\cos }^2\theta {\sin }^2\theta$.

It is $(2\sin \theta \cos \theta {)}^2$, which is exactly ${\sin }^{2}2\theta$. So the expression becomes $\cos 2\theta (1-{\sin }^{2}2\theta )$.

And what is $1-{\sin }^{2}2\theta$? It is ${\cos }^{2}2\theta$! We have arrived at ${\cos }^{3}2\theta$.

The entire, terrifying expression has collapsed into a single, elegant term. Finally, we substitute $\theta =\frac{\pi }{8}$ back in, giving us ${\cos }^3\frac{\pi }{4}$.

Since $\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}$, our final answer is:

This, my friend, is the joy of mathematics. We started with a complex, intimidating expression and, through the systematic application of identities and algebraic intuition, reduced it to a simple, beautiful constant. Keep this mindset, and no problem will ever be too difficult for you.