(cos20°)^4+(cos40°)^4+(cos80°)^4
=(cos20°)^4+(cos40°)^4+(sin10°)^4
=1/4*((cos40°+1)^2+(cos80°+1)^2+(1-cos20°)^2)
=1/4*(1+2cos40°+(cos40°)^2+1+2sin10°+(sin10°)^2+1-2cos20°+(cos20°)^2)
=1/4*(3+2cos40°+(1+cos80°)/2+2cos80°+(1-cos20°)/2-2cos20°+(1+cos40°)/2)
=1/4*(3+3/2+5/2*(cos40°+cos80°-cos20°))
=1/4*(9/2+5/2*(sin50°+sin10°-cos20°))
=1/4*(9/2+5/2*(2sin30°cos20°-cos20°))
=1/4*(9/2)
=9/8