We'll use
rarra^3+b^3=(a+b)(a^2-ab+b^2)
rarra^2+b^2=(a-b)^2+2ab
rarrsin^2x+cos^2x=1
rarr2cos^2x=1+cos2x and
rarr2sin^2x=1-cos2x
LHS=cos^6(x)+sin^6(x)
=(cos^2x)^3+(sin^2x)^3
=[cos^2x+sin^2x][(cos^2x)^2-cos^2x*sin^2x+sin^2x)^2]
=1*[(cos^2x-sin^2x)^2+2cos^2x*sin^2x-cos^2x*sin^2x]
=[cos^2(2x)+cos^2x*sin^2x]
=1/4[4cos^2(2x)+4cos^2x*sin^2x]
=1/4[2(1+cos4x)+sin^2(2x)]
=2/(4*2)[2+2cos4x+sin^2(2x)]
=1/8[4+4cos4x+2sin^2(2x)]
=1/8[4+4cos4x+1-cos4x]
=1/8[5+3cos4x]=RHS