Animated Solution for Mathematics - Trigonometry: The value of $\sin 10^{\circ }\sin 30^{\circ }\sin 50^{\circ }\sin 70^{\circ }$ is :-

The Art of Pattern Recognition in Trigonometry. Welcome, future engineer. Today, we are not just solving a math problem; we are learning to see the hidden architecture of trigonometry.

When you look at the expression $E=\sin 10^{\circ }\sin 30^{\circ }\sin 50^{\circ }\sin 70^{\circ }$, your first instinct might be to reach for a calculator or start expanding terms using complex addition formulas. Stop. Take a breath.

In the JEE Advanced arena, the most elegant solutions are rarely the ones that require brute force. They are the ones that require observation.

The Low-Hanging Fruit

Look at the expression again. Do you see it? Among the four terms, one stands out like a beacon: $\sin 30^{\circ }$.

This is a standard angle, a value etched into the memory of every serious student. We know that $\sin 30^{\circ }=\frac{1}{2}$.

By isolating this term, we immediately simplify our expression to:

We have already reduced the complexity of the problem by 25 percent. This is the first rule of competitive exams: always simplify the knowns before tackling the unknowns.

The Hidden Symmetry

Now, look at the remaining terms: $\sin 10^{\circ }$, $\sin 50^{\circ }$, and $\sin 70^{\circ }$. They seem random, but they are not. They are dancing around the number $60^{\circ }$.

Notice that $50^{\circ }=60^{\circ }-10^{\circ }$ and $70^{\circ }=60^{\circ }+10^{\circ }$. This is not a coincidence; it is a mathematical signature.

Whenever you see a product of sines where the angles are of the form $\theta$, $60^{\circ }-\theta$, and $60^{\circ }+\theta$, you must immediately recall the powerful identity:

This identity is a shortcut to victory. It collapses a product of three terms into a single sine function.

The Grand Unification

Let us apply this to our problem. By setting $\theta =10^{\circ }$, our expression $\sin 10^{\circ }\sin 50^{\circ }\sin 70^{\circ }$ maps perfectly onto the identity.

It becomes $\frac{1}{4}\sin (3\cdot 10^{\circ })$, which simplifies to $\frac{1}{4}\sin 30^{\circ }$.

Now, bring back the $\frac{1}{2}$ we set aside earlier. Our expression $E$ is now:

This is the beauty of mathematics—the way complex terms cancel out and simplify into something manageable.

The Final Victory

We are at the finish line. We have $E=\frac{1}{8}\sin 30^{\circ }$.

Since we know $\sin 30^{\circ }=\frac{1}{2}$, we substitute it one last time:

There it is. The answer is $\frac{1}{16}$. It is clean, it is precise, and it is the result of recognizing a pattern rather than fighting the algebra.

Remember this journey. The next time you see a product of trigonometric ratios, do not panic. Look for the symmetry, look for the $60^{\circ }$ relationship, and let the identity do the heavy lifting for you. You have the tools; now go out and conquer.