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If $y = \log_e \tan \left(\frac{\pi}{4} + \frac{x}{2}\right)$,then $\tanh \left(\frac{y}{2}\right) = $

$\tan \left(-\frac{23 \pi}{3}\right)-\cot \left(\theta-\frac{13 \pi}{3}\right) =$

If $\tan \theta = \frac{a}{b},$ then $\frac{\sin \theta}{\cos^8 \theta} + \frac{\cos \theta}{\sin^8 \theta} = $

Let $\theta$ be an angle in the standard position such that the point $(-5, 12)$ lies on its terminal side. Then:

Which of the following have the same value?
$(a)$ $\sin 120^{\circ}$
$(b)$ $\cos 930^{\circ}$
$(c)$ $\tan 840^{\circ}$
$(d)$ $\cot (-1110^{\circ})$

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