If $\sec 4 A =\operatorname{cosec}\left( A -20^{\circ}\right),$ where $4 A$ is an acute angle, find the value of $A$. (in $^{\circ}$)

In $\triangle$ $PQR,$ right-angled at $Q$ (see $Fig.$), $PQ =3 \,cm$ and $PR =6 \,cm$. Determine $\angle QPR$ and $\angle PRQ$.

Show that:

$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ}=1$

$(ii)$ $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}=0$

Express the ratios $\cos A ,$ tan $A$ and $\sec A$ in terms of $\sin A .$

Evaluate:

$\operatorname{cosec} 31^{\circ}-\sec 59^{\circ}$

Evaluate:

$\frac{\sin 18^{\circ}}{\cos 72^{\circ}}$