The value of cosec$$^{2}\ 18^\circ\ - \frac{1}{cot^{2}72^\circ}\ $$is

cosec$$^{2}\ 18^\circ\ - \frac{1}{cot^{2}72^\circ}\ $$

= cosec$$^{2} 18^\circ - tan^{2} 72^\circ$$ ($$\because \frac{1}{cot^{2} \ominus}$$=$$tan^{2}\ominus$$)

= cosec$$^{2} 18^\circ$$ - $$tan^{2} (90-72)^\circ$$
= cosec$$^{2} 18^\circ$$ - $$sec^{2} 18^\circ$$ ($$\because sec^{2}\ominus= tan^{2}(90^\circ-\ominus)$$)
cosec$$^{2} 18^\circ$$ - $$sec^{2} 18^\circ$$=1($$\because cosec^{2} \ominus$$ - $$sec^{2} \ominus$$=1)