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Verify the following identity:

[tex]\[
\cos ^4 \frac{\pi}{8} + \cos ^4 \frac{3 \pi}{8} + \sin ^4 \frac{5 \pi}{8} + \sin ^4 \frac{7 \pi}{8} = \frac{3}{2}
\][/tex]

Sagot :

GinnyAnswer
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].
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