Question

Найдите значение выражения

Answer 1

$\sin2\alpha\cos5\alpha-\sin\alpha\cos6\alpha=2\sin\alpha\cos\alpha\cos5\alpha-\sin\alpha\cos6\alpha=\\ \\ =\sin \alpha(2\cos \alpha\cos5\alpha-\cos6\alpha)=\sin\alpha(\cos 4\alpha+\cos6\alpha-\cos6\alpha)=\\ \\ =\sin\alpha\cos4\alpha=\sin\alpha(2\cos^22\alpha-1)=\sin\alpha\Big(2\cdot (1-2\sin^2\alpha)^2-1\Big)=\\ \\ =\sin\alpha(2-8\sin^2\alpha+8\sin^4\alpha-1)=a\Big(8a^4-8a^2+1\Big)$

$\cos7\alpha\cos4\alpha-\cos8\alpha\cos3\alpha=\dfrac{1}{2}\Big(\cos3\alpha+\cos11\alpha\Big)-\dfrac{1}{2}\Big(\cos 5\alpha+\\ \\ +\cos11\alpha\Big)=\dfrac{1}{2}\Big(\cos3\alpha-\cos5\alpha\Big)=\dfrac{1}{2}\cdot 2\sin\dfrac{3\alpha+5\alpha}{2}\sin\dfrac{5\alpha-3\alpha}{2}\\ \\ =\sin4\alpha\sin\alpha=2\sin2\alpha\cos2\alpha\sin\alpha=4\sin^2\alpha\cos\alpha(2\cos^2\alpha-1)\\ \\ =4(1-\cos^2\alpha)\cos\alpha(2\cos^2\alpha-1)=4a(1-a^2)(2a^2-1)$