Question

Sin30*ctg30-cos30cos^2L*tg^2L+sin^2L*ctg^2L1+sin(П+2)*cos(3П/2+L)(1-sin30)*(1+sin30)+(1+cos30)(1-cos30)

Answer 1

1). $sin 30^{\circ}*ctg 30^{\circ} - cos 30^{\circ} = sin 30^{\circ}* \frac{cos 30^{\circ}}{sin 30^{\circ}} - cos 30^{\circ} =$
$=cos 30^{\circ} - cos 30^{\circ} = 0$

2). $cos^{2} \alpha *tg^{2} \alpha +sin^{2} \alpha *ctg^{2} \alpha =sin^{2} \alpha +cos^{2} \alpha =1$

3). $1+sin( \pi + \alpha )*cos(\frac{3 \pi }{2}+ \alpha) =1-sin \alpha *sin \alpha =cos^{2} \alpha$

4). $(1-sin30^{\circ})(1+sin30^{\circ}) +(1+cos30^{\circ})(1-cos30^{\circ}) =$
$= 1-sin^{2}30^{\circ}+1-cos^{2} 30^{\circ} = 2-1=1$

Answer 2

Давайте посчитаем выражение по частям:

1. Раскроем тригонометрические функции: - sin(30) = 0.5 - cos(30) = √3/2 - ctg(30) = 1/tan(30) = 1/√3 - sin^2(L) = (1 - cos(2L))/2 - cos^2(L) = (1 + cos(2L))/2 - ctg^2(L) = 1/tan^2(L) = 1/(sin^2(L)/cos^2(L)) = cos^2(L)/sin^2(L) - sin(П + 2) = sin(П)cos(2) + cos(П)sin(2) = -sin(2) = -2sin(1)cos(1) - cos(3П/2 + L) = cos(3П/2)cos(L) - sin(3П/2)sin(L) = 0 - (-1)sin(L) = sin(L) - sin(П) = 0 - cos(П) = -1 - sin(2) = 2sin(1)cos(1) - cos(2) = cos^2(1) - sin^2(1) = 1 - 2sin^2(1) - sin(3П/2) = -1 - cos(3П/2) = 0

2. Подставим значения в выражение:

Sin30*ctg30 - cos30*cos^2L*tg^2L + sin^2L*ctg^2L1 + sin(П+2)*cos(3П/2+L)(1-sin30)*(1+sin30) + (1+cos30)(1-cos30)

= 0.5 * (1/√3) - (√3/2) * ((1 + cos(2L))/2) * (cos^2(L)/sin^2(L)) + ((1 - cos(2L))/2) * (1/√3)^2 + (-2sin(1)cos(1)) * sin(L) * (1 - 0.5) * (1 + 0.5) + (1 + (√3/2)) * (1 - (√3/2))

= 0.5/√3 - (√3/2) * (1 + cos(2L))/2 * (cos^2(L)/sin^2(L)) + (1 - cos(2L))/2 * (1/3) + (-2sin(1)cos(1)) * sin(L) * 0.5 * 1.5 + (1 + (√3/2)) * (1 - (√3/2))

3. Упростим выражение:

= 0.5/√3 - (√3/2) * (1 + cos(2L))/2 * (cos^2(L)/sin^2(L)) + (1 - cos(2L))/6 + (-2sin(1)cos(1)) * sin(L) * 0.75 + (1 + (√3/2)) * (1 - (√3/2))

= 0.5/√3 - (√3/2) * (1 + cos(2L))/2 * (cos^2(L)/sin^2(L)) + (1 - cos(2L))/6 - (3sin(1)cos(1))/2 * sin(L) + (1 + (√3/2)) * (1 - (√3/2))

4. Продолжим упрощение:

= 0.5/√3 - (√3/2) * (1 + cos(2L))/2 * (cos^2(L)/sin^2(L)) + (1 - cos(2L))/6 - (3sin(1)cos(1))/2 * sin(L) + 1 - (√3/2)

= 0.5/√3 - (√3/2) * (1 + cos(2L))/2 * (cos^2(L)/sin^2(L)) + (1 - cos(2L))/6 - (3sin