There is no solution to the equation sec(x) = 0.
A. True
B. False
The correct answer and explanation is:
The correct answer is A. True.
To explain why, we need to understand the behavior of the secant function. The secant function, sec(x)\sec(x), is the reciprocal of the cosine function, so: sec(x)=1cos(x)\sec(x) = \frac{1}{\cos(x)}
For sec(x)\sec(x) to be zero, the reciprocal of cos(x)\cos(x) would have to be zero, which implies that cos(x)\cos(x) would have to be infinite. However, the cosine function can never be infinite; it only ranges from -1 to 1. Thus, there is no value of xx for which sec(x)=0\sec(x) = 0.
Mathematically, this can be shown by recognizing that for the secant function to equal zero, the following equation would have to hold: 1cos(x)=0\frac{1}{\cos(x)} = 0
This implies: cos(x)=∞\cos(x) = \infty
But, as stated earlier, cosine can never be infinite. Therefore, there is no solution to the equation sec(x)=0\sec(x) = 0, making the statement true.
In summary, the secant function can never be zero because its reciprocal, the cosine function, cannot take the value of infinity. Therefore, the equation sec(x)=0\sec(x) = 0 has no solution.