Question

已知tanx=2,求cos^4(π/3+x)-cos^4(x-π/6)的值

Answer 1

tanx=2 平方得
sin²x=4cos²x
5cos²x=1
cos²x=1/5 sin²x=4/5
sin2x=2sinxcosx=4/5
cos2x=2cos²x-1=-3/5
cos^4(π/3+x)-cos^4(x-π/6)
=cos^4(π/3+x)-cos^4(π/6-x)
=cos^4(π/3+x)-sin^4(π/3+x)
=cos^2(π/3+x)-sin^2(π/3+x)
=cos(2x+2π/3)
=(-1/2)cos2x-(√3/2)sin2x
=(-1/2)(-3/5)-(√3/2)(4/5)
=(3-4√3)/10