Prove: \cos {}^(6) ( \alpha ) + \sin {}^(6) ( \alpha ) = (1)/(4) (1 + 3 \cos {}^(2) (2 \alpha ) = (1)/(8) (5 + 3 \cos(4 \alpha ) )

Question

Prove:


`\cos {}^(6) ( \alpha ) + \sin {}^(6) ( \alpha ) = (1)/(4) (1 + 3 \cos {}^(2) (2 \alpha ) = (1)/(8) (5 + 3 \cos(4 \alpha ) )`

Answer 1

Use the double angle identity:


`\cos^2\alpha=\frac{1+\cos2\alpha}2`


`\sin^2\alpha=\frac{1-\cos2\alpha}2`

This lets us write


`\cos^6\alpha+\sin^6\alpha=\left(\frac{1+\cos2\alpha}2\right)^3+\left(\frac{1-\cos2\alpha}2\right)^3`


`=\frac{1+3\cos2\alpha+3\cos^22\alpha+\cos^32\alpha}8+\frac{1-3\cos2\alpha+3\cos^22\alpha-\cos^32\alpha}8`


`=\frac{2+6\cos^22\alpha}8=\frac{1+3\cos^22\alpha}4`

Use the identity again to write


`=\frac{1+3\frac{1+\cos4\alpha}2}4`


`=\frac{5+3\cos4\alpha}8`