Animated Solution for Mathematics - Trigonometry: Without using tables, prove that $(\sin 12^{\circ })(\sin 48^{\circ })(\sin 54^{\circ })=1/8$.

A Journey to $1/8$

Imagine you are standing on the edge of a complex trigonometric landscape. You are faced with the product $(\sin 12^{\circ })(\sin 48^{\circ })(\sin 54^{\circ })$, and your goal is to prove it equals $1/8$.

At first glance, these angles—$12^{\circ }$, $48^{\circ }$, and $54^{\circ }$—seem like strangers. They do not belong to the standard $30^{\circ }$, $45^{\circ }$, or $60^{\circ }$ family. But in the world of JEE Advanced, there are no strangers, only hidden connections waiting to be revealed.

The Art of Strategic Grouping

When you see a product of sines, your first instinct should be to look for a way to break the product into a sum. We utilize the identity:

If we look at $48^{\circ }$ and $12^{\circ }$, their difference is $36^{\circ }$ and their sum is $60^{\circ }$. Both are incredibly useful, marking the 'spark' of the problem—the realization that $48^{\circ }$ and $12^{\circ }$ are meant to be paired together.

The Transformation

We apply our tool to the first two terms:

Substituting the known values, we obtain:

Now, we bring back the third term, $\sin 54^{\circ }$, which has been waiting patiently. Our expression is now:

The Complementary Bridge

We are almost there, but we have a mix of $36^{\circ }$ and $54^{\circ }$. This is where the complementary angle identity saves the day.

Since $\sin 54^{\circ }=\cos (90^{\circ }-54^{\circ })=\cos 36^{\circ }$, the entire expression is unified under the angle $36^{\circ }$:

The Final Algebraic Dance

Expanding the expression, we get:

Using the special value $\cos 36^{\circ }=\frac{\sqrt{5}+1}{4}$, we find its square:

Substituting these values back into our expression:

The $\sqrt{5}$ terms cancel out, leaving us with $\frac{2}{16}$, which simplifies to the final result of $1/8$.