Trignometry - Cos

Question

If $\cos54 = t$, determine $\cos144$ in terms of $t$?

What I've done so far is $\cos54=t$, $\cos144 = 2t-18$? Quite lost with this equation, should I have a look at some identities, and is my answer completely incorrect?

Answer 1

We have that

$$\cos 144=\cos(90+54)=\cos 90\cos 54-\sin 90\sin54=-\sin 54.$$ Now, $$\sin 54 =\sqrt{1-\cos^2 54}=\sqrt{1-t^2}.$$ So, the answer is

$$\cos 144=-\sqrt{1-t^2}.$$