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cot.alpha+cot(60+alpha)-cot(60-alpha)=3cot3alpha

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see

http://www.askiitians.com/forums/Trigonometry/prove-cot-a-cot-60-a-cot-60-a-3-cot3a_106221.htm

for one solution. You might try proving it without recourse to sin and cos.

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