А) cos15 cos75 = cos15 cos(90-15) = cos15 sin15 = 2sin15 cos15 =
2
= sin(2*15) = sin30 = 1 = 1/4
2 2 2*2
б) (cos(π/8) + sin(π/8)) (cos³(π/8) - sin³(π/8)) =
=(cos(π/8) + sin(π/8))(cos(π/8) - sin(π/8))(cos²(π/8)+cos(π/8) *sin(π/8)+sin²(π/8))= =(cos²(π/8) - sin²(π/8)) (1 + 2sin(π/8) cos(π/8)) =
2
= cos(2 * (π/8)) * (1 + sin(2 * (π/8))) =
2
= cos(π/4) * (1 + sin(π/4)) =
2
= √2 ( 1 + √2 ) = √2 (4 + √2) = 4√2 + 2 = 2 (2√2 + 1) = 2√2 +1
2 2*2 2 4 2*4 2*4 4
в) 1+tgα tg2α = 1
cos2α
1+tgα tg2α = 1 + sinα * sin2α = 1 + sinα * 2sinα cosα =
cosα cos2α cosα cos2α
= 1 + 2sin²α = cos2α + 2sin²α = cos²α - sin²α + 2sin²α =
cos2α cos2α cos2α
= cos²α + sin²α = 1
cos2α cos2α
1 = 1
cos2α cos2α
Что и требовалось доказать.