Question

Упростите выражения пожалуйста:
1)sin(-альфа)/cos^2(-альфа)-1
2)sin^2(-альфа)-1/cos(-альфа)
3sin^2альфа-1+cos^2альфа+(1-sinальфа)(1+sinальфа)
4)2sin^2альфа-1/1-2cos^2альфа
5)cos^2альфа-1/3-3sin^2альфа
6)4/sin^2альфа-4 ctg^2альфа

Answer 1

$1) frac{sin(- alpha )}{cos ^{2}(- alpha ) - 1} = frac{-sin alpha }{cos ^{2} alpha - 1 } = frac{-sin alpha }{-sin ^{2} alpha } = frac{1}{ sin alpha } = sin ^{-1} alpha$

$2) frac{sin ^{2}(- alpha )-1 }{cos(- alpha )} = frac{sin ^{2} alpha - 1 }{cos alpha } = frac{-cos^{2} alpha }{cos alpha } = -cos alpha$

$3) sin ^{2} alpha - 1 + cos ^{2} alpha + (1 - sin ^{2} alpha )(1 + sin ^{2} alpha ) =$ $-cos ^{2} alpha + cos ^{2} alpha +1 - sin ^{2} alpha = cos ^{2} alpha$

$4) frac{2sin ^{2} alpha - 1 }{1 - 2cos ^{2} alpha } = frac{- cos ^{2 } alpha + sin ^{2} alpha }{sin ^{2} alpha - cos ^{2} alpha } = 1$

$5) frac{cos ^{2} alpha - 1 }{3 - 3 sin^{2} alpha } = frac{-sin ^{2} alpha }{3cos ^{2} alpha } = -frac{1}{3} tg ^{2} alpha$

$6) frac{4}{sin ^{2} alpha } - 4ctg ^{2} alpha = frac{4}{sin ^{2} alpha } -frac{4cos ^{2} alpha }{sin ^{2} alpha } = frac{4 - 4cos ^{2} alpha }{sin ^{2} alpha } = frac{4sin ^{2} alpha }{sin ^{2} alpha } = 4$