5. ctg a*sin a - (1 - 2cos^2 a) / (sin a - cos a) = = cos a/sin a*sin a + (2cos^2 a - 1) / (sin a - cos a) = = cos a + (2cos^2 a - 1) / (sin a - cos a) = = cos a + (cos^2 a - sin^2 a) / (sin a - cos a) = = cos a + (cos a - sin a)(cos a + sin a) / (sin a - cos a) = = cos a - (cos a + sin a) = -sin a cos(3pi/2 - a) = -sin a по формулам приведения Обе части равны одному и тому же. Тождество доказано.
6. sin(a - b) + 2sin b*cos a = sin a*cos b - cos a*sin b + 2sin b*cos a = = sin a*cos b + cos a*sin b = sin(a + b) = sin(25 + 5) = sin 30 = 1/2
7. y = 2 - cos x Область определения (-oo, +oo) Множество значений [1, 3] 2 - cos x = 1 cos x = -1 x = pi + 2pi*k
8. sin^2 t = (√5 - 2)^2/9 = (5 - 4√5 + 4)/9 = (9 - 4√5)/9 cos^2 t = 4√5/9 sin^2 t + cos^2 t = (9 - 4√5)/9 + 4√5/9 = 9/9 = 1 Это верно при любом t
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5. ctg a*sin a - (1 - 2cos^2 a) / (sin a - cos a) == cos a/sin a*sin a + (2cos^2 a - 1) / (sin a - cos a) =
= cos a + (2cos^2 a - 1) / (sin a - cos a) =
= cos a + (cos^2 a - sin^2 a) / (sin a - cos a) =
= cos a + (cos a - sin a)(cos a + sin a) / (sin a - cos a) =
= cos a - (cos a + sin a) = -sin a
cos(3pi/2 - a) = -sin a по формулам приведения
Обе части равны одному и тому же. Тождество доказано.
6. sin(a - b) + 2sin b*cos a = sin a*cos b - cos a*sin b + 2sin b*cos a =
= sin a*cos b + cos a*sin b = sin(a + b) = sin(25 + 5) = sin 30 = 1/2
7. y = 2 - cos x
Область определения (-oo, +oo)
Множество значений [1, 3]
2 - cos x = 1
cos x = -1
x = pi + 2pi*k
8. sin^2 t = (√5 - 2)^2/9 = (5 - 4√5 + 4)/9 = (9 - 4√5)/9
cos^2 t = 4√5/9
sin^2 t + cos^2 t = (9 - 4√5)/9 + 4√5/9 = 9/9 = 1
Это верно при любом t