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Solvecos 2 A ...

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Solve
$\frac{c o s^{2} A}{1 - t a n A} + \frac{s i n^{3} A}{s i n A - c o s A} = 1 + s i n A c o s A$

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Solution

$\frac{c o s^{2} A}{1 - t a n A} + \frac{s i n^{3} A}{s i n A - c o s A} = 1 + s i n A c o s A L H S : \frac{c o s^{2} A}{1 - \frac{S i n A}{c o s A}} + \frac{s i n^{3} A}{s i n A - c o s A} = \frac{c o s^{3} A}{c o s A - s i n A} - \frac{s i n^{3} A}{c o s A - s i n A} = \frac{c o s^{3} A - s i n^{3} A}{c o s A - s i n A} = \frac{(c o s A - s i n A) (c o s^{2} A + s i n^{2} A)}{(c o s A - s i n A)}$
= 1 + sin A cos A = R.H.S proved
(Using identity $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ )

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