已知sin(3π+θ)=1/4 求cos(π+θ)/cosθ[cos(π+θ)-1] +cos(θ

已知sin(3π+θ)=1/4
求cos(π+θ)/cosθ[cos(π+θ)-1] +cos(θ-2π)/cos(θ+2π)cos(π+θ)+cos(-θ)的值

sin(3π+θ)=1/4

sin(π+θ)=1/4
-sinθ=1/4
sinθ=-1/4
cosθ=±√(1-sin^2θ)=±√15/4
cos(π+θ)/cosθ[cos(π+θ)-1] +cos(θ-2π)/cos(θ+2π)cos(π+θ)+cos(-θ)

=-cosθ/cosθ[-cosθ-1]+cosθ/cosθ(-cosθ)+cosθ
=cosθ+1-cosθ+cosθ
=cosθ
=±√15/4

sin(3π+θ)=sin(π+θ)=-sinθ=1/4 sinθ=-1/4 cos(π+θ)/cosθ[cos(π+θ)-1] +cos(θ-2π)/cos(θ+2π)cos(π+θ)+cos(-θ) =-cosθ/cosθ[-cosθ-1] +cosθ/cosθ(-cosθ)+cosθ =1/(cosθ+1)-1/cosθ+cosθ

sina+sinb=1/4① cosa+cosb=1/3② ①^2+②^2得 sina^2+sinb^2+cosa^2+cosb^2+2sinasinb+2cosacosb=1/9+1/16=25/144 即2+2cos(a-b)=25/144, ∴cos(a-b)=25/288-1=-263/288 ②^2-①^2得 -sina^2-sinb^2+cosa^2+cosb^2-2sinasinb+2cosacosb=1/9-1/16=7/144 即cos2a+cos2b+2cos(a+b)=7/144 和差化积公式 cos2a+cos2b=2cos(a+b)cos(a-b)=-263cos(a+b)/144 ∴2cos(a+b)-263cos(a+b)/144=(25/144)cos(a+b)=7/144 ∴cos(a+b)=7/25