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Calculate the following without using a calculator:

(a) $2 \sin 15^{\circ} \cdot \cos 15^{\circ}$

(b) $1-2 \cos ^2 22,5^{\circ}$

(c) $\left(\cos 15^{\circ}+\sin 15^{\circ}\right)^2$
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(a)
\begin{aligned} & 2 \sin 15^{\circ} \cdot \cos 15^{\circ} \\ & =\sin \left[2\left(15^{\circ}\right)\right] \\ & =\sin 30^{\circ} \\ & =\frac{1}{2} \end{aligned}
(b)
\begin{aligned} & 1-2 \cos ^2 22,5^{\circ} \\ & =-\left(2 \cos ^2 22,5^{\circ}-1\right) \\ & =-\left[\cos \left(2\left(22,5^{\circ}\right)\right]\right. \\ & =-\cos 45^{\circ} \\ & =-\frac{\sqrt{2}}{2} \end{aligned}

(c) \begin{aligned} & \left(\cos 15^{\circ}+\sin 15^{\circ}\right)^2 \\ = & \cos ^2 15^{\circ}+2 \cos 15^{\circ} \cdot \sin 15^{\circ}+\sin ^2 15^{\circ} \\ = & \cos ^2 15^{\circ}+\sin ^2 15^{\circ}+2 \sin 15^{\circ} \cdot \cos 15^{\circ} \\ = & 1+\sin \left[2\left(15^{\circ}\right)\right] \\ = & 1+\sin 30^{\circ} \\ = & 1+\frac{1}{2}=\frac{3}{2}\end{aligned}
by Diamond (71,587 points)

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