State whether the following statement is true or false. Justify your answer.
$\cot A$ is the product of cot and A.

Hint: The reciprocal of cotangent (cot) is a tangent (tan). The cot A is a cotangent of angle A. There are 6 trigonometric identities. Those are sin, cosine, tangent, cotangent, secant and cosecant.

Complete step-by-step answer:
The cotangent function is an old mathematical function. It was mentioned in 1620 by E. Gunter who invented the notation of cotangens.
In a right triangle, the cotangent of an angle is the length of the adjacent side divided by the length of the opposite side. In a formula, it is abbreviated to just cot. Of the six possible trigonometric functions, cotangent, secant, and cosecant, are rarely used. This is the reciprocal of the tangent function.
They can be easily replaced with derivations of the more common three: sin, cos and tan. Cotangent can be derived in two ways.
$\cot A=\dfrac{1}{\tan A},\cot A=\dfrac{\cos A}{\sin A}$
For every trigonometry function such as cot, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. So the inverse of cot is arccot.
The given statement cot A is the product of cot and A is False.
The cotangent function is the ratio of the length of the adjacent side to that of the opposite side.
$\cot A=\dfrac{1}{\tan A}=\dfrac{\text{side adjacent to angle A}}{\text{side opposite to angle A}}$
The cot A is a single term and cot without A does not have any meaning.


Note: Note that arctan and cot are really separate things - $\cot x=\dfrac{1}{\tan x}$ , so cotangent is basically the reciprocal of a tangent, or, in other words, the multiplicative inverse. The arctan(x) is the angle whose tangent is x.