Question

三角恒等变换问题

设f(x)=cos²(x-θ)-2cosθ*cos(x-θ)*cosx+cos²x.试证明：f(x)与x无关。

Answer 1

f(x)=cos²(x-θ)-2cosθ*cos(x-θ)*cosx+cos²x.
=0.5+0.5cos(2x-2θ)-2cosθ*cos(x-θ)*cosx+0.5+0.5cos2x
=1+0.5[cos(2x-2θ)+cos2x]-2cosθ*cos(x-θ)*cosx
=1+cos(2x-θ)cosθ-2cosθ*cos(x-θ)*cosx
=1+cosθ[cos(2x-θ)-2cos(x-θ)*cosx]
=1+cosθ[cos(2x-θ)-cos(2x-θ)-cosθ]
=1+cos²θ.