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Without using trigonometric tables, prove that
(i) \( \sin ^{2} 12^{\circ}+\sin ^{2} 78^{\circ}=1 \)
(ii) \( \sec ^{2} 29^{\circ}-\cot ^{2} 61^{\circ}=1 \)
(iii) \( \tan ^{2} 56^{\circ}-\cot ^{2} 34^{\circ}=0 \)
(iv) \( \cos ^{2} 57^{\circ}-\sin ^{2} 33^{\circ}=0 \)
(v) \( \sec ^{2} 50^{\circ}-\cot ^{2} 40^{\circ}=1 \)
(vi) \( \operatorname{cosec}^{2} 72^{\circ}-\tan ^{2} 18^{\circ}=1 \)
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