Animated Solution for Mathematics - Trigonometry: The general solution of $\sin x-3\sin 2x+\sin 3x=\cos x-3\cos 2x+\cos 3x$ is

Imagine you are standing before a complex trigonometric equation:

At first glance, it looks like a chaotic jumble of terms. But in the world of JEE Advanced, chaos is often just order waiting to be discovered.

The first step in any great mathematical adventure is to look for patterns. We have angles $x$, $2x$, and $3x$. Notice the symmetry? The angles $x$ and $3x$ are perfectly balanced around $2x$.

By rearranging the equation to group the smallest and largest angles, we get:

This simple act of grouping is not just algebraic manipulation; it is a strategic move to unlock the power of trigonometric identities.

Now that we have grouped our terms, we need a tool to simplify them. We utilize the sum-to-product identities:

By substituting $C=3x$ and $D=x$, the magic happens. The left side transforms into $2\sin 2x\cos x-3\sin 2x$, and the right side becomes $2\cos 2x\cos x-3\cos 2x$.

Suddenly, the complexity collapses. We have converted addition into multiplication, which is the key to factoring.

With our new expression, $\sin 2x(2\cos x-3)=\cos 2x(2\cos x-3)$, we see a common factor: $(2\cos x-3)$. It is tempting to just cancel it out, but pause!

In mathematics, the most dangerous move is to divide by a variable expression without checking if it can be zero. Let us analyze $(2\cos x-3)=0$. This implies $\cos x=1.5$.

But wait, the range of $\cos x$ is strictly $[-1,1]$. It can never reach $1.5$. Because this factor can never be zero, we are mathematically safe to divide both sides by it.

After dividing, we are left with the elegant simplicity of $\sin 2x=\cos 2x$. Dividing by $\cos 2x$ gives us:

We know that $\tan \theta =1$ at $\theta =\frac{\pi }{4}$. Since the tangent function is periodic with $\pi$, the general solution for $2x$ is:

Finally, dividing by $2$ gives us our destination:

We started with a daunting equation and ended with a clean, elegant solution. This is the beauty of trigonometry—it rewards those who look for symmetry and respect the constraints of the functions.