Question

sin^4 pi/8 +sin^4 3pi/8 + sin^4 5pi/8 + sin^4 7pi/8

Answer 1

Sin^4 pi/8 +sin^4 3pi/8 +sin^4 5pi/8 + sin^4 7pi/8=3/2

Answer 2

$\bold{\sin ^{4}\left(\frac{\pi}{8}\right)+\sin ^{4}\left(\frac{3 \pi}{8}\right)+\sin ^{4}\left(\frac{5 \pi}{8}\right)+\sin ^{4}\left(\frac{7 \pi}{8}\right)=\frac{3}{2}}$

Solution:

Let us use the trigonometric identity $(\sin x)=\sin (\pi-x)$ using this identity let us convert

$\sin ^{4}\left(\frac{5 \pi}{8}\right)+\sin ^{4}\left(\frac{7 \pi}{8}\right) \text { into } \sin ^{4}\left(\frac{3 \pi}{8}\right)+\sin ^{4}\left(\frac{\pi}{8}\right)$

Putting the value $\sin ^{4}\left(\frac{3 \pi}{8}\right)+\sin ^{4}\left(\frac{\pi}{8}\right) back\ into\ \sin ^{4}\left(\frac{\pi}{8}\right)+\sin ^{4}\left(\frac{3 \pi}{8}\right)+\sin ^{4}\left(\frac{5 \pi}{8}\right)+\sin ^{4}\left(\frac{7 \pi}{8}\right)$

We get $\sin ^{4}\left(\frac{\pi}{8}\right)+\sin ^{4}\left(\frac{3 \pi}{8}\right)+\sin ^{4}\left(\frac{3 \pi}{8}\right)+\sin ^{4}\left(\frac{\pi}{8}\right)$

Adding them and finding common and using the trigonometric identity $\sin \left(\frac{\pi}{2}-x\right)=\cos x$

So $\sin ^{4}\left(\frac{3 \pi}{8}\right)=\sin ^{4}\left(\frac{\pi}{2}-\frac{3 \pi}{8}\right)=\cos ^{4}\left(\frac{\pi}{8}\right)$

$2\left(\sin ^{4}\left(\frac{\pi}{8}\right)+\sin ^{4}\left(\frac{3 \pi}{8}\right)\right)=2\left(\sin ^{4}\left(\frac{\pi}{8}\right)+\cos ^{4}\left(\frac{\pi}{8}\right)\right)$

Using the formula of $(a+b)^{2}=a^{2}+b^{2}+2 a b$ we get

$2\left(\left(\sin ^{2}\left(\frac{\pi}{8}\right)\right)^{2}+\left(\cos ^{2}\left(\frac{\pi}{8}\right)\right)^{2}=\left(\sin ^{2}\left(\frac{\pi}{8}\right)\right)+\left(\cos ^{2}\left(\frac{\pi}{8}\right)\right)^{2}-2 \sin ^{2}\left(\frac{\pi}{8}\right) \cos ^{2}\left(\frac{\pi}{8}\right)\right)$

$2\left(\sin ^{2}\left(\frac{\pi}{8}\right)^{2}+\cos ^{2}\left(\frac{\pi}{8}\right)^{2}=1-2 \sin ^{2}\left(\frac{\pi}{8}\right) \cos ^{2}\left(\frac{\pi}{8}\right)\right)$

$2\left(1-\frac{\left(\sin ^{2}\left(\frac{\pi}{4}\right)\right)^{2}}{2}\right)$;    using the formula sin 2x = 2 sinx . cosx

$2\left(1-\frac{\left(\sin ^{2}\left(\frac{\pi}{4}\right)\right)^{2}}{2}\right)=2\left(1-\frac{1}{4}\right)=\frac{3}{2}$