Let's find the value of [tex]\(\cos^2 45^\circ - \sin^2 15^\circ\)[/tex] step by step.
1. Convert angles from degrees to radians:
- [tex]\(45^\circ\)[/tex] in radians is [tex]\(\frac{45 \cdot \pi}{180} = \frac{\pi}{4} \approx 0.7853981633974483\)[/tex] radians.
- [tex]\(15^\circ\)[/tex] in radians is [tex]\(\frac{15 \cdot \pi}{180} = \frac{\pi}{12} \approx 0.2617993877991494\)[/tex] radians.
2. Compute the trigonometric values:
- [tex]\(\cos 45^\circ = \cos\left(\frac{\pi}{4}\right) \approx 0.7071067811865476\)[/tex].
- [tex]\(\sin 15^\circ = \sin\left(\frac{\pi}{12}\right) \approx 0.25881904510252074\)[/tex].
3. Square the trigonometric values:
- [tex]\(\cos^2 45^\circ = \left(0.7071067811865476\right)^2 = 0.5000000000000001\)[/tex].
- [tex]\(\sin^2 15^\circ = \left(0.25881904510252074\right)^2 = 0.06698729810778066\)[/tex].
4. Calculate the difference:
[tex]\[
\cos^2 45^\circ - \sin^2 15^\circ = 0.5000000000000001 - 0.06698729810778066 = 0.43301270189221946
\][/tex]
Therefore, the value of [tex]\(\cos^2 45^\circ - \sin^2 15^\circ\)[/tex] is approximately [tex]\(\boxed{0.43301270189221946}\)[/tex].
