AMU 2026 — Mathematics PYQ

1. Simplify Using Trigonometric Identity

The given equation features an expression on the left-hand side that directly matches the standard tangent subtraction formula:

$$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$

By setting $A=3x$ and $B=2x$, the equation simplifies significantly:

$$\tan (3x-2x)=1$$

$$\tan x=1$$

2. Find the General Solution

The principal solution where $\tan x=1$ occurs at $x=\frac{\pi }{4}$. Therefore, the standard general solution for the tangent function is given by:

$$x=n\pi +\frac{\pi }{4},\text{where }n\in \mathbb{Z}$$

3. Check the Domain and Undefined Points (Crucial Step)

Before finalizing the solution, any values of $x$ that make the individual components of the original equation undefined must be excluded. The original expression contains terms for $\tan 3x$ and $\tan 2x$.

The tangent function $\tan \theta$ becomes undefined when its argument is an odd multiple of $\frac{\pi }{2}$:

$$\theta \ne (2k+1)\frac{\pi }{2},k\in \mathbb{Z}$$

Let's test our general solution $x=n\pi +\frac{\pi }{4}$ for the term $\tan 2x$:

Substitute $x=n\pi +\frac{\pi }{4}$ into $2x$:

$$2x=2\left(n\pi +\frac{\pi }{4}\right)$$

$$2x=2n\pi +\frac{\pi }{2}$$

$$2x=(4n+1)\frac{\pi }{2}$$

Since $(4n+1)$ is always an odd integer for any integer value of $n$, the angle $2x$ will always land exactly on an odd multiple of $\frac{\pi }{2}$.

Consequently:

$$\tan 2x=\tan \left((4n+1)\frac{\pi }{2}\right)=\pm \infty \text{(Undefined)}$$

Conclusion

Because the expression $\tan 2x$ is completely undefined for every single value produced by our general solution formula, there are no permissible values of $x$ that can satisfy the original mathematical statement.

Thus, the solution set is empty, which is denoted by the null set symbol $\varphi$.

This directly matches option (a).