Animated Solution for Mathematics - Trigonometry: The number of elements in the set $\{x\in [0,180^{\circ }]:\tan (x+100^{\circ })=\tan (x+50^{\circ })\tan x\tan (x-50^{\circ })\}$ is .........

Imagine you are standing on the edge of a complex trigonometric landscape. You are presented with the equation:

At first glance, it looks like a chaotic mess of tangents. However, notice the rhythm: the angles $x-50^{\circ }$, $x$, and $x+50^{\circ }$ are in an arithmetic progression. This is a hidden invitation to use the power of symmetry.

To tame this equation, we must first bring order to the chaos. By rearranging the terms, we obtain:

This ratio form is our first major breakthrough. Now, we convert everything into the fundamental language of trigonometry: sines and cosines.

The left side becomes:

The right side becomes:

Here is where the JEE Advanced spirit truly shines. We apply the Componendo and Dividendo rule: if $\frac{a}{b}=\frac{c}{d}$, then $\frac{a+b}{a-b}=\frac{c+d}{c-d}$.

Applying this to our equation, the left side numerator becomes $\sin (x+100^{\circ })\cos x+\cos (x+100^{\circ })\sin x$, which is the classic expansion for $\sin (A+B)$. The denominator becomes $\sin (x+100^{\circ })\cos x-\cos (x+100^{\circ })\sin x$, which is $\sin (A-B)$.

On the right side, the terms collapse into $\cos (A-B)$ and $\cos (A+B)$. The equation simplifies beautifully to:

With the equation now in the form $\sin (4x+100^{\circ })+\cos 50^{\circ }=0$, we are on the home stretch. We convert $\cos 50^{\circ }$ to $\sin 40^{\circ }$, and then rewrite the equation as $\sin (4x+100^{\circ })=\sin (-40^{\circ })$.

Using the general solution:

We can systematically find the values of $x$ in the range $[0,180^{\circ }]$. By testing $n=1,2,3,4$, we find the solutions:

There are exactly four solutions. You have navigated the complexity, applied the right tools, and arrived at the truth. That is the essence of JEE Advanced mathematics.