Express cos12a in terms of cos3a and in terms of sin3a

Question

Express  cos12a in terms of cos3a and in terms of sin3a

anonymous · Answer

So far in terms of cos3a, I got 2(cos^23a-1)^2-1
and
in terms of sin3a:
1-2(2sin3acos3a)^2

anonymous · Answer

@ganeshie8 @zepdrix

ganeshie8 · Answer

$$\begin{align}\cos(12a)&=\cos(2\cdot 6a)\&=2\cos^2(6a)-1\&=2[\cos(2\cdot 3a)]^2-1\&=2[\color{red}{2}\cos^2(3a)-1]^2-1\end{align}$$

ganeshie8 · Answer

right ?

anonymous · Answer

Oh I see. I wrote the identity wrong. Oops

ganeshie8 · Answer

for sin(3a) part, you may start with above result and simply replace $\cos^2(3a)$ by $1-\sin^2(3a)$

ganeshie8 · Answer

cos(12a) in terms of sin(3a) : $$\begin{align}\cos(12a)&=\cos(2\cdot 6a)\&=2\cos^2(6a)-1\&=2[\cos(2\cdot 3a)]^2-1\&=2[1-\color{red}{2}\sin^2(3a)]^2-1\end{align}$$

ganeshie8 · Answer

you need to know the double angle identity : 
$$\cos(2t) ~=~ 2\cos^2(t)-1~=~1-2\sin^2t$$

anonymous · Answer

Thank you @ganeshie8! I got it now! :D