Final answer:
Sec x = k and cosec x = k have no solutions when the value of k is between -1 and 1 because this falls outside the range of these reciprocal trigonometric functions.
Step-by-step explanation:
When sec x = k and cosec x = k have no solutions, it means that the value of k falls outside the range of these trigonometric functions. For secant (sec x), which is the reciprocal of the cosine function, it will have no solution when k is between -1 and 1, that is, -1 < k < 1, since the cosine function has values ranging from -1 to 1, and the reciprocal of a number between -1 and 1 would not be a real number.
Similarly, for cosecant (cosec x), which is the reciprocal of the sine function, it will have no solution when k also lies between -1 and 1, since the sine function's range is also between -1 and 1.
When you have the equations sec(x) = k and cosec(x) = k, there are no solutions when k has an absolute value greater than or equal to 1.
To see why, let's look at the graphs of sec(x) and cosec(x). The graph of sec(x) is the reciprocal of the graph of cos(x), and the graph of cosec(x) is the reciprocal of the graph of sin(x).
Both sin(x) and cos(x) have values between -1 and 1. So, if k is greater than or equal to 1 or less than or equal to -1, there will be no values of x where sec(x) or cosec(x) equals k.