Question

Without using trigonometric tables, evaluate ${\sin }^2{34}^{\circ }+{\sin }^2{56}^{\circ }+2\tan {18}^{\circ }\tan {72}^{\circ }-{\cot }^2{30}^{\circ }$

Answer 1

Given:
${\sin }^2{34}^{\circ }+{\sin }^2{56}^{\circ }+2\tan {18}^{\circ }\tan {72}^{\circ }-{\cot }^2{30}^{\circ }$

$={\sin }^2{34}^{\circ }+{\sin }^2({90}^{\circ }-{34}^{\circ })+2\tan {18}^{\circ }\tan ({90}^{\circ }-{18}^{\circ })-{\cot }^2{30}^{\circ }$

$={\sin }^2{34}^{\circ }+{\cos }^2{34}^{\circ }+2\tan {18}^{\circ }\cot {18}^{\circ }-(\sqrt{3}{)}^2$

$=1+2\tan {18}^{\circ }\times \frac{1}{\tan {18}^{\circ }}-3$

$=1+2-3$

$=0$