Question
Without using trigonometrical tables, evaluate:$\operatorname{cosec} 257^{\circ}-\tan ^2 33^{\circ}+\cos 44^{\circ} \operatorname{cosec} 46^{\circ}-\sqrt{2} \cos 45^{\circ}-\tan ^2 60^{\circ}$
✓
Answer
$\cos e c^2 57^{\circ}-\tan ^2 33^{\circ}+\cos 44^{\circ} \cos e c 46^{\circ}-\sqrt{2} \cos 45^{\circ}-\tan ^2 60^{\circ} $
$ =\cos e c^2\left(90^{\circ}-33^{\circ}\right)^{\circ}-\tan ^2 33^{\circ}+\cos 44^{\circ} \cos e c\left(90^{\circ}-44^{\circ}\right)-\sqrt{2} \cos 45^{\circ}-\tan ^2 60^{\circ}$
$=\sec ^2 33^{\circ}-\tan ^2 33^{\circ}+\cos 44^{\circ} \sec 44^{\circ}-\sqrt{2} \cos 45^{\circ}-\tan ^2 60 $
$ =1+1-\sqrt{2} \cos 45^{\circ}-\tan ^2 60$
$ =1+1-\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)-(\sqrt{3})^2 $
$=2-1-3 $
$ =-2$
$ =\cos e c^2\left(90^{\circ}-33^{\circ}\right)^{\circ}-\tan ^2 33^{\circ}+\cos 44^{\circ} \cos e c\left(90^{\circ}-44^{\circ}\right)-\sqrt{2} \cos 45^{\circ}-\tan ^2 60^{\circ}$
$=\sec ^2 33^{\circ}-\tan ^2 33^{\circ}+\cos 44^{\circ} \sec 44^{\circ}-\sqrt{2} \cos 45^{\circ}-\tan ^2 60 $
$ =1+1-\sqrt{2} \cos 45^{\circ}-\tan ^2 60$
$ =1+1-\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)-(\sqrt{3})^2 $
$=2-1-3 $
$ =-2$
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