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

Welcome, future engineers. Today, we are going to dismantle a problem that, at first glance, looks like a nightmare of radicals and powers.

You see ${\cos }^3\frac{\pi }{8}$ and ${\sin }^3\frac{\pi }{8}$, and your instinct might be to reach for the half-angle formulas. You might be tempted to calculate $\cos \frac{\pi }{8}$ as $\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}}$ and then cube it. Stop. Take a breath.

In the JEE Advanced arena, if you find yourself doing heavy arithmetic, you are likely missing the elegant path. Let us learn to see the structure, not just the numbers.

The Power of Abstraction

When you see an angle like $\frac{\pi }{8}$ appearing alongside $\frac{3\pi }{8}$, your brain should immediately recognize a relationship. Let $A=\frac{\pi }{8}$. Then, $3A=\frac{3\pi }{8}$.

Suddenly, the expression transforms from a terrifying collection of constants into a clean, algebraic structure:

This is the first step of a master strategist. By replacing the specific with the general, we strip away the intimidation factor. We are no longer dealing with $\pi$; we are dealing with the relationship between an angle and its triple.

The Triple Angle Toolkit

Now, look at the cubic terms: ${\cos }^{3}A$ and ${\sin }^{3}A$. In trigonometry, whenever you see cubic powers, you should immediately think of the triple-angle identities. They are the only standard tools that bridge the gap between linear angles and cubic powers.

Recall the identities:

Our goal is to isolate the cubic terms. By rearranging these, we get:

Notice how we have converted a 'power' problem into a 'linear' problem. This is the essence of trigonometric simplification—reducing the degree of the expression.

The Substitution and the Dance of Terms

Now, we substitute these back into our original expression. It might look like it's getting longer, but trust the process. The algebra is setting the stage for a beautiful collapse:

Let us factor out the $\frac{1}{4}$ to keep our workspace clean. Expanding the terms, we get:

Look closely at what we have created. We have squared terms and mixed terms. Let us group them strategically:

The Elegant Collapse

This is the moment of truth. Do you recognize the patterns inside the brackets? The first term, ${\cos }^{2}3A-{\sin }^{2}3A$, is the classic double-angle identity for cosine: $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$.

Here, $\theta =3A$, so it becomes $\cos (2\cdot 3A)=\cos 6A$.

The second term, $\cos A\cos 3A+\sin A\sin 3A$, is the expansion of $\cos (x-y)$. Thus, it becomes $\cos (3A-A)=\cos 2A$.

Our expression has collapsed into:

The Final Victory

Now, we bring back our value $A=\frac{\pi }{8}$. Substituting this back into our simplified expression:

We are now in the realm of standard angles. We know that $\cos (\frac{3\pi }{4})=-\frac{1}{\sqrt{2}}$ and $\cos (\frac{\pi }{4})=\frac{1}{\sqrt{2}}$.

Plugging these in:

And there it is. A problem that seemed to require brute-force calculation yielded to the elegance of identities. You didn't just solve a problem; you navigated a logical structure. Keep this mindset—look for the identity, trust the algebra, and the final answer is $\frac{1}{2\sqrt{2}}$.