Question

Докажите тождество:
sin^2*3d / sin^2*d - cos^2*3d / cos^2*d = 8cos2d

Answer 1

$frac{sin ^{2} 3d}{sin ^{2}d } - frac{cos ^{2}3d }{cos ^{2} d} = frac{sin ^{2}3d*cos ^{2}d -cos ^{2} 3d*sin ^{2} d }{sin ^{2} d*cos ^{2} d} = frac{sin(3d-d)sin(3d+d)}{sin ^{2} d*cos ^{2} d} = \ \ = frac{sin2d*sin4d}{sin ^{2} d*cos ^{2} d} = frac{2sindcosd*4sindcosd(cos ^{2} d-sin ^{2} d)}{sin ^{2} d*cos ^{2} d} = \ frac{8sin ^{2}dcos ^{2}d*(cos ^{2}-sin ^{2} d)}{sin ^{2} d*cos ^{2} d} =8(cos ^{2} d-sin ^{2} d)=8cos2d$