**Period of the Function f(x) = cosec²(3x) cot(4x)**

To determine the period of the given function, we need to understand the behavior of the trigonometric functions involved.

1. **Cosecant Function (cosec or csc):**
The cosecant function, denoted as csc(x) or cosec(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). The period of the cosecant function is 2π or 360°.

2. **Cotangent Function (cot):**
The cotangent function, denoted as cot(x), is the reciprocal of the tangent function. It can be represented as cot(x) = 1/tan(x). The period of the cotangent function is also π or 180°.

Now, let's analyze the given function step by step:

1. **f(x) = cosec²(3x) cot(4x)**
- The function consists of two trigonometric functions: cosecant squared (cosec²) and cotangent (cot).
- The argument of the cosecant function is 3x, which indicates that the period of this function will be reduced by a factor of 3.
- Similarly, the argument of the cotangent function is 4x, which means the period of this function will be reduced by a factor of 4.

2. **Period of the Cosecant Squared Function**
- The cosecant squared function, denoted as cosec²(3x), is the square of the cosecant function.
- Since the period of the cosecant function is 2π or 360°, the period of the cosecant squared function will be half of that, which is π or 180°.
- Therefore, the period of the cosecant squared function in this case is π.

3. **Period of the Cotangent Function**
- The cotangent function, denoted as cot(4x), has a period of π or 180°.
- Thus, the period of the cotangent function in this case is π.

4. **Overall Period of the Function f(x)**
- To find the overall period of the given function, we need to determine the least common multiple (LCM) of the periods of the individual functions.
- The LCM of π and π is π.
- Hence, the overall period of the function f(x) = cosec²(3x) cot(4x) is π.

In conclusion, the given function f(x) = cosec²(3x) cot(4x) has a period of π or 180°.