Упростить выражение: (cos(a - pi/4)cos(pi/4 - a))/sin^2(a + pi/4)

Как разобраться с заданием 10 класса: - упростить выражение: (cos(a - pi/4)cos(pi/4 - a))/sin^2(a + pi/4)

 

[Cereswyn]

(Cos(a - π/4) * Cos(π/4 - a))/Sin^2(a + π/4). Cos(a - π/4) = (Cosa * (√2/2) + Sina * (√2/2) = (√2/2) * (Cosa + Sina). Cos(π/4- a) = (√2/2) * Cosa + (√2/2) * Sina = (√2/2) * (Cosa + Sina). (Cos(a - π/4) * Cos(π/4 – a)) = (√2/2) * (Cosa + Sina) * (√2/2) * (Cosa + Sina) = (Cosa + Sina)^2/2. Sin^2(a + π/4) = Sin(a + π/4) * Sin(a + π/4). Sin(a + π/4) = Sina * (√2/2) + Cosa * (√2/2) = (√2/2) * (Sina + Cosa). Sin^2(a + π/4) = (Sina + Cosa)^2/2. ((Cosa + Sina)^2/2)/( (Sina + Cosa)^2/2) = 1. Ответ: 1.