Prove that the following identities where the angles involved are acute angles for which the expressions are defined.
$(c o s e c θ - cot θ)^{2} = \frac{1 - cos θ}{1 + cos θ}$

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Solution

$L H S = (c o s e c θ - c o t θ)^{2}$

$= (\frac{1}{s i n θ} - \frac{c o s θ}{s i n θ})^{2}$

$= \frac{(1 - c o s θ)^{2}}{s i n^{2} θ}$

$= \frac{(1 - c o s θ)^{2}}{1 - c o s^{2} θ}$

$= \frac{(1 - c o s θ)^{2}}{(1 + c o s θ) (1 - c o s θ)}$

LHS = $\frac{1 - c o s θ}{1 + c o s θ}$

$∴ L H S = R H S$

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