Айниятро исбот намоед:
\(\sin{4\alpha}-\sin{5\alpha}-\sin{6\alpha}+\sin{7\alpha}=-4\sin{\frac{\alpha}{2}}\sin{\alpha}\sin{\frac{11\alpha}{2}}\)
\(\sin{4\alpha}-\sin{6\alpha}=2\cos{\frac{4+6\alpha}{2}}\sin{\frac{4-6\alpha}{2}}=-2\cos{5\alpha}\sin{\alpha}\)
\(\sin{7\alpha}-\sin{5\alpha}=2\cos{\frac{7+5\alpha}{2}}\sin{\frac{7-5\alpha}{2}}=2\cos{6\alpha}\sin{\alpha}\)
\(\sin{4\alpha}-\sin{5\alpha}-\sin{6\alpha}+\sin{7\alpha}=2\cos{6\alpha}\sin{\alpha}-2\cos{5\alpha}\sin{\alpha}=\)
\(=2\sin{\alpha}(\cos{6\alpha}-\cos{5\alpha})\)
\(\cos{6\alpha}-\cos{5\alpha}=-2\sin{\frac{6+5\alpha}{2}}\sin{\frac{6-5\alpha}{2}}=-2\sin{\frac{11\alpha}{2}}\sin{\frac{\alpha}{2}}\)
\(\sin{4\alpha}-\sin{5\alpha}-\sin{6\alpha}+\sin{7\alpha}=-2\sin{\frac{11\alpha}{2}}\sin{\frac{\alpha}{2}}\cdot2\sin{\alpha}=\)
\(=-4\sin{\frac{\alpha}{2}}\sin{\alpha}\sin{\frac{11\alpha}{2}}\)
\(\sin{4\alpha}-\sin{5\alpha}-\sin{6\alpha}+\sin{7\alpha}=-4\sin{\frac{\alpha}{2}}\sin{\alpha}\sin{\frac{11\alpha}{2}}\)
Айният исбот шуд.