已知sin^2 30° + sin^2 90° + sin^2 150° = 3/2

sin^2 5° + sin^2 65° + sin^2 125° = 3/2. 通过观察上述两等式的规律已知sin^2 30° + sin^2 90° + sin^2 150° = 3/，请你写出一般性的命题，并加以证明. 我想要十分详细的解释;2

规律：30+60=90 90+60=150
5+60=65 65+60=125
一般性的命题:sin^2X+sin^2(X+60)+sin^2(X+120)=2\3

 sin80=sin(90-10)=cos10 sin²10+sin²80=sin²10+cos²10=1 同理 sin²20+sin²70=1 sin²30+sin²60=1 sin²40+sin²50=1 所以原式=4

(x+60°)+sin²(x+120°) =sin²x+(sinxcos60°+cosxsin60°)²2 证明;x+1/x-根号3/4cos²x+(1/2sinx+根号3/2cosx)²+(-1/4sin²x+根号3/2sinxcosx =3/2sinx+根号3/x+3/x+3/2(sin²x+cos²：sin²x+sin²+(sinxcos120°+cosxsin120°)佩服 =sin²2sinxcosx+1/4sin²x =3/4cos²(x+60°)+sin²(x+120°)=3/：sin²x+sin²2cos²x+3/2cosx)²2sin² =sin²规律