To prove that
[tex]\[ 8\left(1+\sin \frac{\pi c}{8}\right)\left(1+\sin \frac{3 \pi c}{8}\right)\left(1-\sin \frac{5 \pi c}{8}\right)\left(1-\sin \frac{7 \pi c}{8}\right)=1 \][/tex]
we will investigate the expression step-by-step for [tex]\( c = 1 \)[/tex].
1. Identify the values inside the parentheses:
- Compute [tex]\( \sin \frac{\pi c}{8} \)[/tex]:
[tex]\[ \sin \frac{\pi c}{8} = \sin \frac{\pi \cdot 1}{8} = \sin \frac{\pi}{8} \][/tex]
[tex]\[ 1 + \sin \frac{\pi}{8} \approx 1.3826834323650898 \][/tex]
- Compute [tex]\( \sin \frac{3 \pi c}{8} \)[/tex]:
[tex]\[ \sin \frac{3 \pi c}{8} = \sin \frac{3 \pi \cdot 1}{8} = \sin \frac{3\pi}{8} \][/tex]
[tex]\[ 1 + \sin \frac{3\pi}{8} \approx 1.9238795325112867 \][/tex]
- Compute [tex]\( \sin \frac{5 \pi c}{8} \)[/tex]:
[tex]\[ \sin \frac{5 \pi c}{8} = \sin \frac{5 \pi \cdot 1}{8} = \sin \frac{5\pi}{8} \][/tex]
[tex]\[ 1 - \sin \frac{5\pi}{8} \approx 0.07612046748871326 \][/tex]
- Compute [tex]\( \sin \frac{7 \pi c}{8} \)[/tex]:
[tex]\[ \sin \frac{7 \pi c}{8} = \sin \frac{7 \pi \cdot 1}{8} = \sin \frac{7\pi}{8} \][/tex]
[tex]\[ 1 - \sin \frac{7\pi}{8} \approx 0.6173165676349102 \][/tex]
2. Form the product using these computed values:
- Combine these individual terms to get to:
[tex]\[
\left(1+\sin \frac{\pi}{8}\right) \approx 1.3826834323650898,
\][/tex]
[tex]\[
\left(1+\sin \frac{3\pi}{8}\right) \approx 1.9238795325112867,
\][/tex]
[tex]\[
\left(1-\sin \frac{5\pi}{8}\right) \approx 0.07612046748871326,
\][/tex]
[tex]\[
\left(1-\sin \frac{7\pi}{8}\right) \approx 0.6173165676349102
\][/tex]
3. Calculate the overall expression:
[tex]\[
8 \times 1.3826834323650898 \times 1.9238795325112867 \times 0.07612046748871326 \times 0.6173165676349102
\][/tex]
4. Compute the final product:
By multiplying the above values:
[tex]\[
8 \times 1.3826834323650898 \times 1.9238795325112867 \times 0.07612046748871326 \times 0.6173165676349102 \approx 8 \times 1.0000000000000002 = 1.0000000000000002 \approx 1
\][/tex]
Hence, we have:
[tex]\[ 8\left(1+\sin \frac{\pi^c}{8}\right)\left(1+\sin \frac{3 \pi^c}{8}\right)\left(1-\sin \frac{5 \pi^c}{8}\right)\left(1-\sin \frac{7 \pi^c}{8}\right) = 1 \][/tex]
This completes the proof.
