1) tg 255 - tg 195 = tg (255 - 180) - tg (195 - 180) = tg 75 - tg 15 =
= ctg 15 - tg 15 = cos 15 / sin 15 - sin 15 / cos 15
Найдем sin 15 и cos 15 из формулы половинного угла
sin 15 = √((1 - cos 30) / 2) = √((1 - √3/2)/2) = √(2 - √3) / 2
cos 15 = √((1 + cos 30) / 2) = √((1 + √3/2)/2) = √(2 + √3) / 2
ctg 15 - tg 15 = √(2 + √3) / √(2 - √3) - √(2 - √3) / √(2 + √3) =
= √(2 + √3)^2 / √[(2 - √3)(2 + √3)] - √(2 - √3)^2 / √[(2 - √3)(2 + √3)] =
= (2 + √3) / √(4 - 3) - (2 - √3) / √(4 - 3) = 2 + √3 - 2 + √3 = 2√3
2) sin (3pi/2 - 2arctg (4/3)) = -cos(2arctg (4/3))
arctg(4/3) = x, tg x = 4/3; -cos 2x = 1 - 2cos^2 x
cos^2 x = 1 / (1 + tg^2 x) = 1 / (1 + 16/9) = 9/25
-cos 2x = 1 - 18/25 = (25 - 18)/25 = 7/25
sin (3pi/2 - 2arctg (4/3)) = 7/25