Question
Without using tables, evaluate the following $\sec 45^\circ \sin 45^\circ - \sin 30^\circ \sec 60^\circ .$
✓
Answer
$\sec 45^{\circ} \sin 45^{\circ}-\sin 30^{\circ} \sec 60^{\circ} .$
$ \cos 45^{\circ}=\frac{1}{\sqrt{2}}$
$ \Rightarrow \sec 45^{\circ}=\sqrt{2}$
$ \sin 45^{\circ}=\frac{1}{\sqrt{2}}$
$ \sin 30^{\circ}=\frac{1}{2}$
$ \cos 60^{\circ}=\frac{1}{2}$
$ \Rightarrow \sec 60^{\circ}=2$
$ \sec 45^{\circ} \sin 45^{\circ}-\sin 30^{\circ} \sec 60^{\circ}$
$ =\sqrt{2} \times \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} \times 2$
$ =1-1$
$ =0$
$ \cos 45^{\circ}=\frac{1}{\sqrt{2}}$
$ \Rightarrow \sec 45^{\circ}=\sqrt{2}$
$ \sin 45^{\circ}=\frac{1}{\sqrt{2}}$
$ \sin 30^{\circ}=\frac{1}{2}$
$ \cos 60^{\circ}=\frac{1}{2}$
$ \Rightarrow \sec 60^{\circ}=2$
$ \sec 45^{\circ} \sin 45^{\circ}-\sin 30^{\circ} \sec 60^{\circ}$
$ =\sqrt{2} \times \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} \times 2$
$ =1-1$
$ =0$
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