Чоп кардан
Бахш: Тригонометрия
Миқдори намоиш: 1899
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Айниятро исбот намоед:
\(\cos{\alpha}+\cos{2\alpha}+\cos{6\alpha}+\cos{7\alpha}=4\cos{\frac{\alpha}{2}}\cos{\frac{5\alpha}{2}}\cos{4\alpha}\)
\(\cos{\alpha}+\cos{7\alpha}=2\cos{\frac{\alpha+7\alpha}{2}}\cos{\frac{\alpha-7\alpha}{2}}=2\cos{4\alpha}\cos{3\alpha}\)
\(\cos{2\alpha}+\cos{6\alpha}=2\cos{\frac{2\alpha+6\alpha}{2}}\cos{\frac{2\alpha-6\alpha}{2}}=2\cos{4\alpha}\cos{2\alpha}\)
\(\cos{\alpha}+\cos{2\alpha}+\cos{6\alpha}+\cos{7\alpha}=2\cos{4\alpha}\cos{2\alpha}+2\cos{4\alpha}\cos{3\alpha}=\)
\(=2\cos{4\alpha}(\cos{2\alpha}+\cos{3\alpha})\)
\(\cos{2\alpha}+\cos{3\alpha}=2\cos{\frac{2\alpha+3\alpha}{2}}\cos{\frac{2\alpha-3\alpha}{2}}=2\cos{\frac{5\alpha}{2}}\cos{\frac{\alpha}{2}}\)
\(\cos{\alpha}+\cos{2\alpha}+\cos{6\alpha}+\cos{7\alpha}=2\cos{4\alpha}\cdot2\cos{\frac{5\alpha}{2}}\cos{\frac{\alpha}{2}}=\)
\(=4\cos{\frac{\alpha}{2}}\cos{\frac{5\alpha}{2}}\cos{4\alpha}\)
\(\cos{\alpha}+\cos{2\alpha}+\cos{6\alpha}+\cos{7\alpha}=4\cos{\frac{\alpha}{2}}\cos{\frac{5\alpha}{2}}\cos{4\alpha}\)
Айният исбот шуд.