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Sagot :
Answer:
0
Keys:
When going over functions like this, we must use these cosine rules:
- [tex]\cos \left(s\right)+\cos \left(t\right)=2\cos \left(\frac{s+t}{2}\right)\cos \left(\frac{s-t}{2}\right)[/tex]
- [tex]\cos \left(-x\right)=\cos \left(x\right)[/tex]
- [tex]\cos \left(\frac{\pi }{2}\right)=0[/tex]
Step-by-step explanation:
[tex]=\cos \left(\frac{\pi }{5}\right)+2\cos \left(\frac{2\cdot \frac{\pi }{5}+3\cdot \frac{\pi }{5}}{2}\right)\cos \left(\frac{2\cdot \frac{\pi }{5}-3\cdot \frac{\pi }{5}}{2}\right)+\cos \left(4\cdot \frac{\pi }{5}\right)\\=\cos \left(\frac{\pi }{5}\right)+2\cos \left(\frac{\pi }{2}\right)\cos \left(-\frac{\pi }{10}\right)+\cos \left(\frac{4\pi }{5}\right)\\=\cos \left(\frac{\pi }{5}\right)+2\cos \left(\frac{\pi }{2}\right)\cos \left(\frac{\pi }{10}\right)+\cos \left(\frac{4\pi }{5}\right)[/tex]
[tex]cos\left(\frac{\pi }{5}\right) = \frac{\sqrt{5} + 1}{4}\\=\frac{\sqrt{5}+1}{4}+2\cdot \:0\cdot \frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}-\frac{1+\sqrt{5}}{4}\\=0[/tex]
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