Question

40 БАЛЛОВ! Упростите,используя формулу понижения степени cos^2(3П/8-a)-cos^2(11П/8+a)

Answer 1

$cos^2(\frac{3\pi}{8}-\alpha )-cos^2(\frac{11\pi}{8}+\alpha )=\\\\=\frac{1+cos(\frac{3\pi}{4} -2\alpha )}{2}-\frac{1+cos(\frac{11\pi }{4}+2\alpha )}{2}=\\\\=\frac{1}{2}\cdot (cos(\frac{3\pi }{4}-2\alpha )-cos(\frac{11\pi }{4}+2\alpha ))=\frac{1}{2}\cdot 2sin\frac{7\pi}{4}\cdot sin(\pi +2\alpha )=\\\\=sin(2\pi -\frac{\pi}{4})\cdot (-sin2\alpha )=-sin(-\frac{\pi}{4})\cdot sin2\alpha =\frac{\sqrt2}{2}\cdot sin2\alpha$