NIMCET 2024 — Mathematics PYQ

To find the general value of $x-y$, we can use standard trigonometric transformations.

Step 1: Simplify the equations using transformation formulas

We are given two equations:

1. $\sin x-\sin y=0$

2. $\cos x-\cos y=0$

Applying the product-to-sum identities:

• For the sine equation:

  $$2\sin \left(\frac{x-y}{2}\right)\cos \left(\frac{x+y}{2}\right)=0\text{— (Equation 1)}$$

• For the cosine equation:

  $$-2\sin \left(\frac{x-y}{2}\right)\sin \left(\frac{x+y}{2}\right)=0\text{— (Equation 2)}$$

Step 2: Find the common solution

For both Equation 1 and Equation 2 to be simultaneously true, the common factor must be zero.

Notice that $\sin \left(\frac{x-y}{2}\right)$ appears in both equations. If we set the other factors to zero, namely $\cos \left(\frac{x+y}{2}\right)=0$ and $\sin \left(\frac{x+y}{2}\right)=0$, it creates a contradiction because sine and cosine of the same angle cannot equal zero at the same time (${\sin }^2\theta +{\cos }^2\theta =1$).

Therefore, the only valid shared solution is:

$$\sin \left(\frac{x-y}{2}\right)=0$$

Step 3: Determine the general solution

The general solution for $\sin \theta =0$ is $\theta =n\pi$, where $n$ is any integer ($n\in \mathbb{Z}$).

Substituting $\theta =\frac{x-y}{2}$:

$$\frac{x-y}{2}=n\pi$$

Multiply both sides by $2$:

$$x-y=2n\pi$$

Correct Answer: D) $2n\pi$