Question
Evaluate the following:
$(\cos0^\circ+\sin45^\circ+\sin30^\circ)(\sin90^\circ-\cos45^\circ+\cos60^\circ)$
$(\cos0^\circ+\sin45^\circ+\sin30^\circ)(\sin90^\circ-\cos45^\circ+\cos60^\circ)$
✓
Answer
$(\cos0^\circ+\sin45^\circ+\sin30^\circ)(\sin90^\circ-\cos45^\circ+\cos60^\circ)\dots(\text{i})$ By trigonometric ration we have $\cos0^\circ=1,\ \sin45^\circ=\frac{1}{\sqrt{2}},\ \sin30=\frac{1}{2},$$\sin90^\circ=1,\cos45^\circ=\frac{1}{\sqrt{2}}\cos60^\circ=\frac{1}{2}$
By substituting above values in (i), we get $\Big(1+\frac{1}{\sqrt{2}}+\frac{1}{2}\Big)\Big(1-\frac{1}{\sqrt{2}}+\frac{1}{2}\Big)$ $\Big[\frac{3}{2}+\frac{1}{\sqrt{2}}\Big]\Big[\frac{3}{2}-\frac{1}{\sqrt{2}}\Big]$ $\Rightarrow\Big[\frac{3}{2}\Big]^2-\Big[\frac{1}{\sqrt{2}}\Big]=\frac{9}{4}-\frac{1}{2}=\frac{7}{4}$
By substituting above values in (i), we get $\Big(1+\frac{1}{\sqrt{2}}+\frac{1}{2}\Big)\Big(1-\frac{1}{\sqrt{2}}+\frac{1}{2}\Big)$ $\Big[\frac{3}{2}+\frac{1}{\sqrt{2}}\Big]\Big[\frac{3}{2}-\frac{1}{\sqrt{2}}\Big]$ $\Rightarrow\Big[\frac{3}{2}\Big]^2-\Big[\frac{1}{\sqrt{2}}\Big]=\frac{9}{4}-\frac{1}{2}=\frac{7}{4}$
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