Evaluate each of the following

cos $60^{\circ}$ cos $45^{\circ}$ - sin $60^{\circ}$ sin $45^{\circ}$

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Solution

cos $60^{\circ}$ = $\frac{1}{2}$

cos $45^{\circ}$ = $\frac{1}{\sqrt{2}}$

sin $60^{\circ}$ = $\frac{\sqrt{3}}{2}$

sin $45^{\circ}$ = $\frac{1}{\sqrt{2}}$

Now,

cos $60^{\circ}$ cos $45^{\circ}$ - sin $60^{\circ}$ sin $45^{\circ}$

= $\frac{1}{2}$ $\times$ $\frac{1}{\sqrt{2}}$ - $\frac{\sqrt{3}}{2}$ $\times$ $\frac{1}{\sqrt{2}}$

= $\frac{1}{2 \sqrt{2}}$ - $\frac{\sqrt{3}}{2 \sqrt{2}}$

= $\frac{1 - \sqrt{3}}{2 \sqrt{2}}$

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Evaluate each of the following

$(c o s 0^{o} + s i n 45^{o} + s i n 30^{o}) (s i n 90^{o} - c o s 45^{o} + c o s 60^{o})$

s i n 60^{\circ} c o s 30^{\circ} + s i n 30^{\circ} c o s 60^{\circ}

\frac{c o s 45^{\circ}}{s e c 30^{\circ} + c o s e c 30^{\circ}}

sin 60^{\circ} cos 45^{\circ} + cos 60^{\circ} sin 45^{\circ} = \frac{\sqrt{3 + 1}}{2 \sqrt{2}}

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