Animated Solution for Mathematics - Trigonometry: The equation $y=\sin x\sin (x+2)-{\sin }^2(x+1)$ represents a straight line lying in :

Imagine you are standing before a complex, tangled web of trigonometric functions: $y=\sin x\sin (x+2)-{\sin }^2(x+1)$. At first glance, it looks like a nightmare of oscillating waves, a function that might dance wildly across the Cartesian plane.

But in the world of JEE Advanced, appearances are often a veil. Today, we are going to peel back that veil to reveal the elegant, static truth hiding underneath.

When you see products of sines and squares of sines, your mind should immediately race to your trigonometric identity toolkit. We want to simplify this.

The most powerful move we can make is to multiply the entire equation by $2$. Why? Because $2\sin A\sin B$ and $2{\sin }^2\theta$ are the keys to unlocking the door.

By writing $2y=2\sin x\sin (x+2)-2{\sin }^2(x+1)$, we prepare the stage for the Product-to-Sum and Power Reduction identities.

Let us tackle the first term: $2\sin x\sin (x+2)$. Using the identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$, we set $A=x$ and $B=x+2$.

The magic happens instantly: the difference $A-B$ becomes $x-(x+2)=-2$, and the sum $A+B$ becomes $2x+2$. Since $\cos (-\theta )=\cos \theta$, our term simplifies beautifully to $\cos (2)-\cos (2x+2)$.

Now, look at the second term: $2{\sin }^2(x+1)$. We use the power reduction identity $2{\sin }^2\theta =1-\cos (2\theta )$.

By setting $\theta =x+1$, we get $1-\cos (2(x+1))$, which is simply $1-\cos (2x+2)$.

Now, let us bring these pieces together. Substitute them back into our equation for $2y$:

Take a deep breath and look at what happens when we distribute that negative sign. The term $-\cos (2x+2)$ meets the $+\cos (2x+2)$ from the second bracket. They vanish!

They cancel each other out completely, leaving behind only constants:

We are left with $y=\frac{\cos 2-1}{2}$. This is not a function of $x$ at all! It is a constant.

Since $\cos (2)$ is approximately $-0.416$, the value of $y$ is roughly $\frac{-0.416-1}{2}=-0.708$. This is a negative constant.

In the language of geometry, $y=k$ (where $k<0$) is a horizontal line lying strictly below the x-axis.

Because the line exists for all real $x$, it stretches infinitely to the left and right. It occupies the space where $y$ is negative, which corresponds exactly to the third quadrant (where $x$ is negative) and the fourth quadrant (where $x$ is positive).

It never touches the first or second quadrants because those regions require $y$ to be positive.

And there you have it. What started as a terrifying trigonometric expression collapsed into a simple, horizontal line. This is the elegance of mathematics—the ability to find order in chaos.