Question

1)sin^4альфа +cos^4альфа
2)sin^6 альфа +-cos^6 альфа

Answer 1

$1)\ \ sin^4a+cos^4a=(sin^4a+2\cdot sin^2a\cdot cos^2a+cos^4a)-2\cdot sin^2a\cdot cos^2a==(\underbrace{sin^2a+cos^2a}_{1})^2-2\cdot sin^2a\cdot cos^2a=1-2\cdot (sina\cdot cosa)^2==1-2\cdot \Big(\dfrac{1}{2}\cdot sin2a\Big)^2=1-\dfrac{1}{2}\cdot sin^22a$

$2)\ \ sin^6a+cos^6a=(sin^2a)^3+(cos^2a)^3==(\underbrace{sin^2a+cos^2a}_{1})(sin^4a-sin^2a\cdot cos^2a+cos^4a)==(sin^4a+cos^4a)-sin^2a\cdot cos^2a=\Big(1-\dfrac{1}{2}\cdot sin^22a\Big)-(sina\cdot cosa)^2==\Big(1-\dfrac{1}{2}\cdot sin^22a\Big)-\Big(\dfrac{1}{2}\cdot sin2a\Big)^2=1-\dfrac{1}{2}\cdot sin^22a-\dfrac{1}{4}\cdot sin^22a==1-\dfrac{3}{4}\cdot sin^22a$

$3)\ \ sin^6a-cos^6a=(sin^2a)^3-(cos^2a)^3==(\underbrace{sin^2a-cos^2a}_{-cos2a})(sin^4a+sin^2a\cdot cos^2a+cos^4a)==-cos2a\cdot \Big((sin^4a+cos^4a)+sin^2a\cdot cos^2a\Big)==-cos2a\cdot \Big(\Big(1-\dfrac{1}{2}\cdot sin^22a\Big)+(sina\cdot cosa)^2\Big)==-cos2a\cdot \Big(\Big(1-\dfrac{1}{2}\cdot sin^22a\Big)+\Big(\dfrac{1}{2}\cdot sin2a\Big)^2\Big)==-cos2a\cdot \Big(1-\dfrac{1}{2}\cdot sin^22a+\dfrac{1}{4}\cdot sin^22a\Big)==cos2a\cdot \Big(\dfrac{1}{4}\cdot sin^22a-1\Big)$

Формулы:

$sin2a=2\, sina\cdot cosacos2a=cos^2a-sin^2aa^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$