To solve the expression [tex]\(\cos^4 \frac{\pi}{8} + \cos^4 \frac{3 \pi}{8} + \sin^4 \frac{5 \pi}{8} + \sin^4 \frac{7 \pi}{8}\)[/tex], we need to evaluate each term separately and then sum the results.
1. Evaluating [tex]\(\cos^4 \frac{\pi}{8}\)[/tex]:
[tex]\[
\cos^4 \frac{\pi}{8} = (\cos \frac{\pi}{8})^4
\][/tex]
Using the given trigonometric value:
[tex]\[
\cos^4 \frac{\pi}{8} \approx 0.7285533905932737
\][/tex]
2. Evaluating [tex]\(\cos^4 \frac{3\pi}{8}\)[/tex]:
[tex]\[
\cos^4 \frac{3\pi}{8} = (\cos \frac{3\pi}{8})^4
\][/tex]
Using the given trigonometric value:
[tex]\[
\cos^4 \frac{3\pi}{8} \approx 0.021446609406726252
\][/tex]
3. Evaluating [tex]\(\sin^4 \frac{5\pi}{8}\)[/tex]:
Using the identity [tex]\(\sin \theta = \cos (\pi/2 - \theta)\)[/tex], we have:
[tex]\[
\sin^4 \frac{5\pi}{8} = (\sin \frac{5\pi}{8})^4 = (\cos \left(\frac{\pi}{2} - \frac{5\pi}{8}\right))^4 = (\cos \left(\frac{4\pi}{8} - \frac{5\pi}{8}\right))^4 = (\cos \left(-\frac{\pi}{8}\right))^4 = (\cos \frac{\pi}{8})^4
\][/tex]
So:
[tex]\[
\sin^4 \frac{5\pi}{8} \approx 0.7285533905932737
\][/tex]
4. Evaluating [tex]\(\sin^4 \frac{7\pi}{8}\)[/tex]:
Again using the identity [tex]\(\sin \theta = \cos (\pi/2 - \theta)\)[/tex]:
[tex]\[
\sin^4 \frac{7\pi}{8} = (\sin \frac{7\pi}{8})^4 = (\cos \left(\frac{\pi}{2} - \frac{7\pi}{8}\right))^4 = (\cos \left(\frac{4\pi}{8} - \frac{7\pi}{8}\right))^4 = (\cos \left(-\frac{3\pi}{8}\right))^4 = (\cos \frac{3\pi}{8})^4
\][/tex]
So:
[tex]\[
\sin^4 \frac{7\pi}{8} \approx 0.021446609406726266
\][/tex]
5. Summing the Values:
Now, we sum all the evaluated terms:
[tex]\[
\cos^4 \frac{\pi}{8} + \cos^4 \frac{3 \pi}{8} + \sin^4 \frac{5 \pi}{8} + \sin^4 \frac{7 \pi}{8} \approx 0.7285533905932737 + 0.021446609406726252 + 0.7285533905932737 + 0.021446609406726266
\][/tex]
Adding these together gives:
[tex]\[
0.7285533905932737 + 0.021446609406726252 + 0.7285533905932737 + 0.021446609406726266 \approx 1.5
\][/tex]
Therefore, the sum [tex]\(\cos^4 \frac{\pi}{8} + \cos^4 \frac{3 \pi}{8} + \sin^4 \frac{5 \pi}{8} + \sin^4 \frac{7 \pi}{8}\)[/tex] equals [tex]\(\boxed{\frac{3}{2}}\)[/tex].
