wykaż tożsamość

[mateeusz94]

\(\displaystyle{ cos \frac{2 \pi }{7} +cos \frac{4 \pi }{7} +cos \frac{6 \pi }{7} =- \frac{1}{2}

tg^{2} \frac{ \pi }{5} \cdot tg^{2} \frac{2 \pi }{5} =5}\)

z góry dziękuję za pomoc.

[stuart clark]

\(\displaystyle{ = \frac{1}{2. sin\left(\frac{\pi}{7}\right)}\left\{2\; sin\left(\frac{\pi}{7}\right).cos\left(\frac{2\pi}{7}\right)+2\; sin\left(\frac{\pi}{7}\right).cos\left(\frac{4\pi}{7}\right)+2\; sin\left(\frac{\pi}{7}\right).cos\left(\frac{6\pi}{7}\right)\right\}}\)


Teraz za pomocą wzoru \(\displaystyle{ 2\; sin\;A.cos\; B = sin\left(A+B\right)+sin\left(A-B\right)}\)

\(\displaystyle{ = \frac{1}{2. sin\left(\frac{\pi}{7}\right)}\left\{sin\left(\frac{3\pi}{7}\right)-sin\left(\frac{\pi}{7}\right)+sin\left(\frac{5\pi}{7}\right)-sin\left(\frac{3\pi}{7}\right)+sin\left(\frac{7\pi}{7}\right)-sin\left(\frac{5\pi}{7}\right)\right\}}}\)

\(\displaystyle{ = -\frac{sin\left(\frac{\pi}{7}\right)}{2.\; sin\left(\frac{\pi}{7}\right)}=-\frac{1}{2}}\)