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