\(x+y=(\pi / 2)\), then range of \(\cos x \cdot \cos y\) is (a) \([-1,1]\) (b) \([0,1]\) (c) \([-(1 / \sqrt{2}),(1 / \sqrt{2})]\) (d) \([-(1 / 2),(1 / 2)]\)

We will first rewrite the given equation to make it easier to work with when finding bounds for the possible values of \(\cos x\) and \(\cos y\). The given equation is \(x + y = \frac{\pi}{2}\). Subtracting \(x\) from both sides, we get: \[y = \frac{\pi}{2} - x\]

Since \(x\) and \(y\) are angles, it is easier to analyze the bounds for each angle. As both \(x\) and \(y\) are angles, their values lie between \(0\) and \(2\pi\). We can now use the equation \(y = \frac{\pi}{2} - x\) to find the bounds for \(y\) as well. Substituting boundary values of \(x\), we get: For \(x = 0\), \(y = \frac{\pi}{2}\). For \(x = 2\pi\), \(y = \frac{\pi}{2} - 2\pi = -\frac{3\pi}{2}\). Therefore, the value of \(y\) lies between \(-\frac{3\pi}{2}\) and \(\frac{\pi}{2}\). From the specified range of values for \(x\) and \(y\), we can find the bounds for the value of \(\cos x\) and \(\cos y\). As \(\cos x\) and \(\cos y\) vary between \(-1\) and \(1\) for any angles \(x\) and \(y\), the possible values of \(\cos x\) and \(\cos y\) are hence between \(-1\) and \(1\).

Since \(\cos x\) and \(\cos y\) are both between \(-1\) and \(1\), we want to determine the minimum and maximum product of their values. - To obtain the minimum product, we need to find the product of the smallest values of \(\cos x\) and \(\cos y\), which would be \((-1) \cdot (-1) = 1\). - To obtain the maximum product, we need to find the product of the largest values of \(\cos x\) and \(\cos y\), which would be \((1) \cdot (1) = 1\). Thus, the range of \(\cos x \cdot \cos y\) is \([1, 1]\). However, this result is not among the given options. Let's analyze the problem once again. We should note that \(\cos x\) and \(\cos y\) need not be simultaneously maximum or minimum. For instance, when \(\cos x = 1\) and \(\cos y = -1\), the product \(\cos x \cdot \cos y\) becomes \((-1)\). Therefore, considering all possible product values, we can now determine the true range of the expression: The range of \(\cos x \cdot \cos y\) is \([-1, 1]\). So, the correct answer is (a) \([-1, 1]\).