Prove that cosecx(secx -1) - cot x(1- cosx) = sinx-sinxcosx

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Solution

$L H S = c o s e c x (s e c x - 1) - c o t x (1 - c o s x) = (\frac{1}{s i n x}) ((\frac{1}{c o s x}) - 1) - (\frac{s i n x}{c o s x}) (1 - c o s x) = (\frac{1 - c o s x}{s i n x c o s x}) - (\frac{c o s x (1 - c o s x)}{s i n x}) = (\frac{1 - c o s x - c o s^{2} x + c o s^{3} x}{s i n x}) = (\frac{(1 - c o s 62 x) - c o s x (1 - c o s^{2} x)}{s i n x}) = (\frac{(1 - c o s^{2} x) (1 - c o s x)}{s i n x}) = (\frac{s i n^{2} x (1 - c o s x)}{s i n x}) = s i n x - s i n x c o s x$

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