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Бахш: Тригонометрия
Миқдори намоиш: 2063
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Айниятро исбот намоед:

\(\cos{2\alpha}-\cos{3\alpha}-\cos{4\alpha}+\cos{5\alpha}=-4\sin{\frac{\alpha}{2}}\sin{\alpha}\cos{\frac{7\alpha}{2}}\)

\(\cos{2\alpha}-\cos{4\alpha}=-2\sin{\frac{2+4\alpha}{2}}\sin{\frac{2-4\alpha}{2}}=2\sin{3\alpha}\sin{\alpha}\)

\(\cos{5\alpha}-\cos{3\alpha}=-2\sin{\frac{5+3\alpha}{2}}\sin{\frac{5-3\alpha}{2}}=-2\sin{4\alpha}\sin{\alpha}\)

\(\cos{2\alpha}-\cos{3\alpha}-\cos{4\alpha}+\cos{5\alpha}=\)
\(=2\sin{3\alpha}\sin{\alpha}-2\sin{4\alpha}\sin{\alpha}=2\sin{\alpha}(\sin{3\alpha}-\sin{4\alpha})\)

\(\sin{3\alpha}-\sin{4\alpha}=2\cos{\frac{3+4\alpha}{2}}\sin{\frac{3-4\alpha}{2}}=-2\cos{\frac{7\alpha}{2}}\sin{\frac{\alpha}{2}}\)

\(\cos{2\alpha}-\cos{3\alpha}-\cos{4\alpha}+\cos{5\alpha}=\)
\(=-2\cos{\frac{7\alpha}{2}}\sin{\frac{\alpha}{2}}\cdot2\sin{\alpha}=-4\sin{\frac{\alpha}{2}}\sin{\alpha}\cos{\frac{7\alpha}{2}}\)

\(\cos{2\alpha}-\cos{3\alpha}-\cos{4\alpha}+\cos{5\alpha}=-4\sin{\frac{\alpha}{2}}\sin{\alpha}\cos{\frac{7\alpha}{2}}\)

Айният исбот шуд.