Докажите тождество: sina+cosa=√2cos(a-π/4)

Question

Докажите тождество:
sina+cosa=√2cos(a-π/4)

Answer 1

Используем формулу cos(x-y) = cosx*cosy+sinx*siny

sinα+cosα = √2cos(α-π/4) = √2*(cosα*cos(π/4)+sinα*sin(π/4)) = √2*(cosα*(√2/2)+sinα*(√2/2)) = √2*(√2/2)(cosα+sinα) = cosα+sinα