Answer :
To determine the exact value of \(\cos 225^\circ \cos 75^\circ\), we start by finding the cosine values for each angle separately.
### Step 1: Evaluate \(\cos 225^\circ\)
The angle \(225^\circ\) lies in the third quadrant where cosine values are negative. It is calculated as:
[tex]\[ 225^\circ = 180^\circ + 45^\circ \][/tex]
Using the cosine addition formula:
[tex]\[ \cos(225^\circ) = \cos(180^\circ + 45^\circ) \][/tex]
Knowing that \(\cos(180^\circ + \theta) = -\cos(\theta)\):
[tex]\[ \cos(225^\circ) = -\cos(45^\circ) \][/tex]
Recall that \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), thus:
[tex]\[ \cos(225^\circ) = -\frac{\sqrt{2}}{2} \][/tex]
### Step 2: Evaluate \(\cos 75^\circ\)
The angle \(75^\circ\) is in the first quadrant where cosine values are positive:
[tex]\[ 75^\circ = 45^\circ + 30^\circ \][/tex]
Using the cosine addition formula:
[tex]\[ \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \][/tex]
Recall that:
[tex]\[ \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \][/tex]
Substitute these values in:
[tex]\[ \cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \][/tex]
[tex]\[ \cos(75^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
### Step 3: Calculate \(\cos 225^\circ \cos 75^\circ\)
Multiply the two cosine values obtained:
[tex]\[ \cos 225^\circ \cos 75^\circ = \left(-\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) \][/tex]
[tex]\[ = -\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
[tex]\[ = -\frac{\sqrt{2}(\sqrt{6} - \sqrt{2})}{8} \][/tex]
[tex]\[ = -\frac{\sqrt{12} - \sqrt{4}}{8} \][/tex]
[tex]\[ = -\frac{2\sqrt{3} - 2}{8} \][/tex]
[tex]\[ = -\frac{2(\sqrt{3} - 1)}{8} = -\frac{\sqrt{3} - 1}{4} \][/tex]
[tex]\[ = \frac{1 - \sqrt{3}}{4} \][/tex]
Comparing this to the given options, we see it matches:
[tex]\[ \boxed{\frac{1 - \sqrt{3}}{4}} \][/tex]
### Step 1: Evaluate \(\cos 225^\circ\)
The angle \(225^\circ\) lies in the third quadrant where cosine values are negative. It is calculated as:
[tex]\[ 225^\circ = 180^\circ + 45^\circ \][/tex]
Using the cosine addition formula:
[tex]\[ \cos(225^\circ) = \cos(180^\circ + 45^\circ) \][/tex]
Knowing that \(\cos(180^\circ + \theta) = -\cos(\theta)\):
[tex]\[ \cos(225^\circ) = -\cos(45^\circ) \][/tex]
Recall that \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), thus:
[tex]\[ \cos(225^\circ) = -\frac{\sqrt{2}}{2} \][/tex]
### Step 2: Evaluate \(\cos 75^\circ\)
The angle \(75^\circ\) is in the first quadrant where cosine values are positive:
[tex]\[ 75^\circ = 45^\circ + 30^\circ \][/tex]
Using the cosine addition formula:
[tex]\[ \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \][/tex]
Recall that:
[tex]\[ \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \][/tex]
Substitute these values in:
[tex]\[ \cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \][/tex]
[tex]\[ \cos(75^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
### Step 3: Calculate \(\cos 225^\circ \cos 75^\circ\)
Multiply the two cosine values obtained:
[tex]\[ \cos 225^\circ \cos 75^\circ = \left(-\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) \][/tex]
[tex]\[ = -\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
[tex]\[ = -\frac{\sqrt{2}(\sqrt{6} - \sqrt{2})}{8} \][/tex]
[tex]\[ = -\frac{\sqrt{12} - \sqrt{4}}{8} \][/tex]
[tex]\[ = -\frac{2\sqrt{3} - 2}{8} \][/tex]
[tex]\[ = -\frac{2(\sqrt{3} - 1)}{8} = -\frac{\sqrt{3} - 1}{4} \][/tex]
[tex]\[ = \frac{1 - \sqrt{3}}{4} \][/tex]
Comparing this to the given options, we see it matches:
[tex]\[ \boxed{\frac{1 - \sqrt{3}}{4}} \][/tex]
