函数f（x）=1-2sin^2 （x+派/8）+2sin（x+派/8）cos（x+派/8）求详细化简

Question

函数f（x）=1-2sin^2 （x+派/8）+2sin（x+派/8）cos（x+派/8）求详细化简

Answer 1

【cos2x=cos²x-sin²x
sin2x=2sinxcosx
sinx+cosx=√2sin(x+π/4) 】
f（x）=1-2sin^2 （x+派/8）+2sin（x+派/8）cos（x+派/8）
=sin²(x+π/8)+cos²(x+π/8)-2sin²(x+π/8)+2sin(x+π/8)cos(x+π/8)
=cos²(x+π/8)-sin²(x+π/8)+sin(2x+π/4)
=cos(2x+π/4)+sin(2x+π/4)
=√2sin(2x+π/4+π/4)
=√2sin(2x+π/2)
=√2sin[π/2-(-2x)]
=√2cos2x