Доказать тождества sin^63x+cos^63x+3sin^23xcos^23x=1
cos^8x-sin^8x=(cos2x(3+cos4x))/4
ctg70°+4cos70°=√3

Пошаговое объяснение:

2 ) cos^8x-sin^8x=(cos2x(3+cos4x))/4 ;

cos^8x-sin^8x= ( cos⁴x - sin⁴x )( cos⁴x +sin⁴x ) = ( cos²x + sin²x ) *                              *( cos²x -sin²x )*( cos⁴x +sin⁴x ) = 1 * cos2x *( cos⁴x +sin⁴x ) =  cos2x * (cos⁴x + +2cos²xsin²x +sin⁴x - 2cos²xsin²x )= cos2x (( cos²x + sin²x )²- 2cos²xsin²x ) =

= cos2x ( 1 - 1/2 (sin2x)²) = cos2x ( 1 - 1/2 * (1 - cos4x )/2 )= (cos2x(3+cos4x))/4 .

Доведено .

3) ctg70°+4cos70°=√3 ;

ctg70°+4cos70° =ctg(90° - 20°) + 4cos( 90°- 20°) = tg20° + 4sin20° =

= sin20°/cos20° + 4sin20° = ( sin20° + 4sin20°cos20°)/cos20° =

=  ( sin20° + 2sin40°)/cos20° = ( sin20° + 2sin(60° - 20°))/cos20° =

= ( sin20° + 2sin60°cos20° - 2sin20°cos60°)/cos20° =(sin20°+√3cos20° -

- sin20°)/cos20° = √3cos20°/cos20° = √3 .Доведено .