Question
Without using trigonometric tables, evaluate.
$4 \tan 60^{\circ} \sec 30^{\circ}+\frac{\sin 31^{\circ} \sec 59^{\circ}+\cot 59^{\circ} \cot 31^{\circ}}{8 \sin ^2 30^{\circ}-\tan ^2 45^{\circ}}$
$\left[\right.$ Hint . Given exp. $\left.=\frac{\sin \left(90^{\circ}-55^{\circ}\right) \cos 55^{\circ}+\cos \left(90^{\circ}-55^{\circ}\right) \sin 55^{\circ}}{\operatorname{cosec}^2\left(90^{\circ}-80^{\circ}\right)^2-\tan ^2 80^{\circ}}\right]$
$4 \tan 60^{\circ} \sec 30^{\circ}+\frac{\sin 31^{\circ} \sec 59^{\circ}+\cot 59^{\circ} \cot 31^{\circ}}{8 \sin ^2 30^{\circ}-\tan ^2 45^{\circ}}$
$\left[\right.$ Hint . Given exp. $\left.=\frac{\sin \left(90^{\circ}-55^{\circ}\right) \cos 55^{\circ}+\cos \left(90^{\circ}-55^{\circ}\right) \sin 55^{\circ}}{\operatorname{cosec}^2\left(90^{\circ}-80^{\circ}\right)^2-\tan ^2 80^{\circ}}\right]$
