已知tanx=—1/3,计算1/2sinxcosx+cos^x

1/2sinxcosx+cos^x
=1/4*sin2x+(1+cos2x )/2
=1/4*sin2x+1/2*cos2x +1/2
=1/4*2tanx/(1+tan^x)+1/2*(1-tan^x)/(1+tan^x)+1/2
=1/2*tanx/(1+tan^x)+1/2*(1-tan^x)/(1+tan^x)+1/2
=1/2*(-1/3)/[1+(-1/3)^2]+1/2*[1-(-1/3)^2]/[1+(-1/3)^2]+1/2
=(-1/6)/(10/9)+1/2*(8/9)/(10/9)+1/2
=-3/20+2/5+1/2
=-3/20+8/20+1/2
=1/4 +1/2
=3/4

tanx=sinx/cosx=-1/3 (1)
sinx=-1/3cosx
原式=1/2*(-1/3)cos^x+cos^x=5/6cos^x
sin^x+cos^x=1,将（1）带入，cos^x=1/10,原式=1/12

sinx/cosx=tanx=-1/3cosx=-3sinx则cos²x=9sin²x因为sin²x+cos²x=1所以sin²x=1/10cos²x=9/10sinxcosx=sinx(-3sinx)=-3sin²x=-3/10所以原式=3/4