Powers of functions
$$ 1. \ \sin^2 α= \frac {(1 − \cos 2α)}{2} $$
$$ 2. \ \sin^3 α= \frac {(3 \sin α – \sin 3α)}{4} $$
$$ 3. \ \sin^4 α= \frac {(\cos 4a – 4 \cos 2α + 3)}{8} $$
$$ 4. \ \sin^5 α= \frac {(10 \sin α − 5 \sin 3α + \sin 5α )}{16} $$
$$ 5. \ \sin^6 α= \frac {(10 – 15 \cos 2α + 6 \cos 4α – \cos 6α)}{32} $$
$$ 6. \ \cos^2 α= \frac {(1 + \cos 2α)}{2} $$
$$ 7. \ \cos^3 α= \frac {(3 \cos α + \cos 3α)}{4} $$
$$ 8. \ \cos^4 α= \frac {(\cos 4α + 4 \cos 2α + 3)}{8} $$
$$ 9. \ \cos^5 α= \frac {(10 \cos α + 5 \sin 3α + \cos 5α)}{16} $$
$$10. \ \cos^6 α= \frac {(10+15 \cos 2α + 6 \cos 4α + \cos 6α)}{32} $$
$$11. \ \tan^2 α= \frac {\sin^2 α}{\cos^2 α}= \frac {(1− \cos 2α)}{(1 + \cos 2α)} $$
$$12. \ \tan^3 α= \frac {\sin^3 α}{\cos^3 α}= \frac {(3 \sin α − \sin 3α)}{(3 \cos α + \cos 3α)} $$