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Use the double angle formulae of cosine to expand the following:

(1) $\cos 6 \mathrm{~A}$

(2) $\cos 80^{\circ}$

(3) $\cos \mathrm{A}$
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(1)
\begin{aligned} & \cos 6 A=\cos [2(3 A)] \\ & \therefore \cos 6 A=\cos ^2 3 A-\sin ^2 3 A \\ & \therefore \cos 6 A=2 \cos ^2 3 A-1 \\ & \therefore \cos 6 A=1-2 \sin ^2 3 A \end{aligned}

(2)
\begin{aligned} & \cos 80^{\circ}=\cos \left[2\left(40^{\circ}\right)\right] \\ & \therefore \cos 80^{\circ}=\cos ^2 40^{\circ}-\sin ^2 40^{\circ} \\ & \therefore \cos 80^{\circ}=2 \cos ^2 40^{\circ}-1 \\ & \therefore \cos 80^{\circ}=1-2 \sin ^2 40^{\circ} \end{aligned}

(3)
\begin{aligned} & \cos A=\cos \left[2\left(\frac{A}{2}\right)\right] \\ & \therefore \cos A=\cos ^2 \frac{A}{2}-\sin ^2 \frac{A}{2} \\ & \therefore \cos A=2 \cos ^2 \frac{A}{2}-1 \\ & \therefore \cos A=1-2 \sin ^2 \frac{A}{2} \end{aligned}
by Diamond (71,587 points)

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