Question

Упростите выражение: 
$cos(450^0-6 alpha )sin^3(180^0-2 alpha )-cos(6 alpha -180^0)sin^3(90^0-2 alpha )$

Answer 1

сos(90-6a)sin³2a+cos6acos³2a=sin6asin³2a+cos6acos³2a
Из формул
sin3a=3cos²asina-sin³a    и cos3a=cos³a-3cosasin²a⇒
sin³2a=3cos²2asin2a-sin6a  и  cos³2a=3cos2asin²2a+cos6a

sin6a(3cos²2asin2a-sin6a )+cos6a(3cos2asin²2a+cos6a)=
=3cos²2asin2asin6a-sin²6a +3cos2asin²2acos6a+cos²6a=
=3cos2asin2a(cos2asin6a+sin2acos6a)+cos12a=3*1/2sin4asin(6a+2a)+cos12a=
=3/2sin4asin8a+cos12a=3/2*1/2(cos(8a-4a)-cos(8a+4a)+cos12a=
=3/4cos4a-3/4cos12a+cos12a=3/4cos4a+1/4cosa12a=3/4cos4a+1/4(4cos³4a-3cos4a)=
=3/4cos4a+cos³4a-3/4cos4a=cos³4a
Из формулы
сos3a=4cos³a-3cosa⇒cos12a=4cos³4a-3cos4a

Answer 2

Спасибо