6. cot a + cot (60° + a) - cot (60°...
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6. cot a + cot (60° + a) - cot (60° – a) = 3 cot 3a.

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Maths
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( begin{aligned} text { Hint }: & text { Use } cot (A+B)=frac{cot A cdot cot B-1}{cot B+cot A} & text { and } cot (A-B)=frac{cot A cdot cot B+1}{cot B-cot A} &=cos u t: cot (A+B)-cot (A-B)=frac{-2 cot B cdot operatorname{cosec}^{2} A}{cot ^{2} B-cot ^{2} A} & cot alpha+cos (60+alpha)-cot (60-alpha) =& cot alpha+left(-frac{2 cot alpha cdot cos e^{2} 60}{cot ^{2} alpha-cot ^{2} 60}right) =& cot alpha quad frac{2 cot alpha times frac{4}{3}}{cot ^{2} alpha-frac{1}{3}} Rightarrow cot alpha-frac{8 cot alpha}{3 cot ^{2} alpha-1}=& frac{3 cot ^{3} alpha-cot alpha-8 cot alpha}{3 cot ^{2} alpha-1}=3 cot (3 alpha) Rightarrow frac{3left(cot ^{3} alpha-3 cot alpharight)}{left(3 cot ^{2} alpha-1right)} end{aligned} )
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