JEE Mathematics 2026 Question with Solution

Given: We need the number of elements in the set $\{x\in[0,180^\circ]: \tan(x+100^\circ)=\tan(x+50^\circ)\tan x\tan(x-50^\circ)\}$.

Find: The number of admissible values of $x$ in $[0,180^\circ]$.

Using the tangent identity for equally spaced angles, the given equation is satisfied for all values of $x$ for which all tangent terms are defined.

So the task reduces to excluding those values of $x$ for which any of the expressions $\tan(x+100^\circ)$, $\tan(x+50^\circ)$, $\tan x$, or $\tan(x-50^\circ)$ is undefined.

Tangent is undefined when its angle is $90^\circ + k\cdot180^\circ$.

Therefore, we exclude values satisfying

$$x=90^\circ+k\cdot180^\circ$$

and similarly values for which

$$x\pm50^\circ=90^\circ+k\cdot180^\circ$$

In the interval $[0,180^\circ]$, the number of excluded values is $30$.

Hence,

$$\text{Total valid values}=180-30=150$$

Therefore, the number of elements in the given set is $150$.

Given: The equation involves only tangent functions at shifted angles.

Find: How many values of $x$ in $[0,180^\circ]$ remain after removing undefined cases.

The solution states that the identity holds generally, so counting solutions means counting all admissible values in the interval.

The only obstruction is that tangent must be defined at every factor appearing in the equation. Thus we check the restrictions coming from each term:

• $\tan x$
• $\tan(x-50^\circ)$
• $\tan(x+50^\circ)$
• $\tan(x+100^\circ)$

Each tangent expression is undefined when its angle is of the form $90^\circ+k\cdot180^\circ$. After excluding all such values lying in $[0,180^\circ]$, the solution gives $30$ excluded values.

So the count of admissible values is

$$180-30=150$$

Therefore, the required number of elements is $150$.