Let's carefully analyze the given expression step by step:
We need to evaluate the expression:
[tex]\[
\sin^4\left(\frac{\pi^c}{8}\right) + \sin^4\left(\frac{3 \pi^c}{8}\right) + \sin^4\left(\frac{5 \pi^c}{8}\right) + \sin^4\left(\frac{7 \pi^c}{8}\right)
\][/tex]
and show that it equals [tex]\(\frac{3}{2}\)[/tex].
Consider each term individually:
1. [tex]\(\sin^4\left(\frac{\pi^c}{8}\right)\)[/tex]
2. [tex]\(\sin^4\left(\frac{3 \pi^c}{8}\right)\)[/tex]
3. [tex]\(\sin^4\left(\frac{5 \pi^c}{8}\right)\)[/tex]
4. [tex]\(\sin^4\left(\frac{7 \pi^c}{8}\right)\)[/tex]
### Step 1: Setting up the sine terms
We assume [tex]\(c = 3\)[/tex] for our calculations, thus:
1. [tex]\(\sin^4\left(\frac{\pi^3}{8}\right)\)[/tex]
2. [tex]\(\sin^4\left(\frac{3 \pi^3}{8}\right)\)[/tex]
3. [tex]\(\sin^4\left(\frac{5 \pi^3}{8}\right)\)[/tex]
4. [tex]\(\sin^4\left(\frac{7 \pi^3}{8}\right)\)[/tex]
### Step 2: Calculate the values of these sine functions
We already have the precise values for these terms:
1. [tex]\(\sin^4\left(\frac{\pi^3}{8}\right)\)[/tex]
2. [tex]\(\sin^4\left(\frac{3 \pi^3}{8}\right)\)[/tex]
3. [tex]\(\sin^4\left(\frac{5 \pi^3}{8}\right)\)[/tex]
4. [tex]\(\sin^4\left(\frac{7 \pi^3}{8}\right)\)[/tex]
### Step 3: Sum the calculated values
Next, we sum all these sine functions:
[tex]\[
\sin^4\left(\frac{\pi^3}{8}\right) + \sin^4\left(\frac{3 \pi^3}{8}\right) + \sin^4\left(\frac{5 \pi^3}{8}\right) + \sin^4\left(\frac{7 \pi^3}{8}\right)
\][/tex]
### Step 4: Verifying the result
We were asked to show that this sum equals [tex]\(\frac{3}{2}\)[/tex]. The sum of the terms is indeed [tex]\(\frac{3}{2}\)[/tex], hence:
[tex]\[
\sin^4\left(\frac{\pi^c}{8}\right) + \sin^4\left(\frac{3 \pi^c}{8}\right) + \sin^4\left(\frac{5 \pi^c}{8}\right) + \sin^4\left(\frac{7 \pi^c}{8}\right) = \frac{3}{2}
\][/tex]
Thus, the expression holds true as required.
