Final answer:
To prove that the expression a⋅cot(x)−b⋅csc(x) has at least one value of c for which f(c) is defined, we need to consider values of x for which sin(x) is not zero.
Step-by-step explanation:
To prove that the expression a⋅cot(x)−b⋅csc(x) has at least one value of c for which f(c) is defined, we can start by understanding the definitions of cotangent and cosecant.
The cotangent of an angle x is defined as the ratio of the adjacent side to the opposite side of a right triangle formed by x. The cosecant of an angle x is defined as the reciprocal of the sine of x, which is equal to 1 divided by the sine of x.
Since the expression involves cot(x) and csc(x), we need to make sure that the angles x we consider are such that the cotangent and cosecant are defined. This means that the sine of x should not be zero, since the sine of an angle is the reciprocal of the cosecant. Therefore, x cannot be any angle for which sin(x) = 0.
In conclusion, as long as we consider values of x for which sin(x) is not zero, the expression a⋅cot(x)−b⋅csc(x) will have at least one value of c for which f(c) is defined.