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nth Term of A.P

cot α + cot ...

Question

$c o t α + c o t (60^{\circ} + α) - c o t (60^{\circ} - α) = 3 c o t 3 α$

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Solution

L.H.S. = $\frac{c o s α}{s i n α} + \frac{c o s (60^{\circ} + α)}{s i n (60^{\circ} + α)} - \frac{c o s (60^{\circ} - α)}{s i n (60^{\circ} - α)}$
$s i n (60^{\circ} - α) c o s (60^{\circ} + α)$
$= \frac{c o s α}{s i n α} + \frac{- c o s (60^{\circ} - α) s i n (60^{\circ} + α)}{s i n^{2} - 60^{\circ} - s i n^{2} α}$
$= \frac{c o s α (\frac{3}{4} - s i n^{2} α) + s i n α s i n (60^{\circ} - α - 60^{\circ} - α)}{s i n α (\frac{3}{4} - s i n^{2} α)}$
$= \frac{c o s α (3 - 4 s i n^{2} α) + 4 s i n α s i n (- 2 α)}{4 s i n α (\frac{3 - 4 s i n^{2} α}{4})}$
$= \frac{c o s a l p h a (3 - 4 s i n^{2} α) - 4 s i n α \cdot s i n α c o s α}{3 s i n α - 4 s i n^{3} α}$
$= \frac{c o s α [3 - 12 (1 - c o s^{2} α)]}{s i n 3 α}$
$= 3 \cdot \frac{[4 c o s^{3} α - 3 c o s a l p h a]}{s i n 3 a l p h a} = 3 \cdot \frac{c o s 3 a l p h a}{s i n 3 a l p h a} = 3 c o t 3 α$

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