Question
Without using trigonometric tables, evaluate
$\sin ^2 34^{\circ}+\sin ^2 56^{\circ}+2 \tan 18^{\circ} \tan 72^{\circ}-\cot ^2 30^{\circ}$
$\sin ^2 34^{\circ}+\sin ^2 56^{\circ}+2 \tan 18^{\circ} \tan 72^{\circ}-\cot ^2 30^{\circ}$
✓
Answer
$\sin ^2 34^{\circ}+\sin ^2 56^{\circ}+2 \tan 18^{\circ} \tan 72^{\circ}-\cot ^2 30^{\circ}:$
$=\sin ^2 34^2+\sin ^2\left(90^{\circ}-34^{\circ}\right)+2 \tan 18^{\circ} \tan \left(90^{\circ}-18^{\circ}\right)-\cot ^2 30^{\circ}$
$=\sin ^2 34^{\circ}+\cos ^2 34^{\circ}+2 \tan 18^{\circ} \cot 18^{\circ}-\cot ^2 30^{\circ}$
$=\left(\sin ^2 34^{\circ}+\cos ^2 34^{\circ}\right)+2 \tan 18^{\circ} \times \frac{1}{\tan 18^{\circ}}-\cot ^2 30^{\circ}$
$=1+2 \times 1-(\sqrt{3})^2 $
$ =1+2-3$
$=3-3 $
$ =0$
$=\sin ^2 34^2+\sin ^2\left(90^{\circ}-34^{\circ}\right)+2 \tan 18^{\circ} \tan \left(90^{\circ}-18^{\circ}\right)-\cot ^2 30^{\circ}$
$=\sin ^2 34^{\circ}+\cos ^2 34^{\circ}+2 \tan 18^{\circ} \cot 18^{\circ}-\cot ^2 30^{\circ}$
$=\left(\sin ^2 34^{\circ}+\cos ^2 34^{\circ}\right)+2 \tan 18^{\circ} \times \frac{1}{\tan 18^{\circ}}-\cot ^2 30^{\circ}$
$=1+2 \times 1-(\sqrt{3})^2 $
$ =1+2-3$
$=3-3 $
$ =0$
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