Use the double angle identity:
[tex]\cos^2\alpha=\dfrac{1+\cos2\alpha}2[/tex]
[tex]\sin^2\alpha=\dfrac{1-\cos2\alpha}2[/tex]
This lets us write
[tex]\cos^6\alpha+\sin^6\alpha=\left(\dfrac{1+\cos2\alpha}2\right)^3+\left(\dfrac{1-\cos2\alpha}2\right)^3[/tex]
[tex]=\dfrac{1+3\cos2\alpha+3\cos^22\alpha+\cos^32\alpha}8+\dfrac{1-3\cos2\alpha+3\cos^22\alpha-\cos^32\alpha}8[/tex]
[tex]=\dfrac{2+6\cos^22\alpha}8=\dfrac{1+3\cos^22\alpha}4[/tex]
Use the identity again to write
[tex]=\dfrac{1+3\frac{1+\cos4\alpha}2}4[/tex]
[tex]=\dfrac{5+3\cos4\alpha}8[/tex]
