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Definition of Functions

Prove that: ...

Question

Prove that: $\frac{c o s e c θ + cot θ}{c o s e c θ - cot θ} = 1 + 2 {cot}^{2} θ + 2 c o s e c θ cot θ$

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Solution

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

$= \frac{c o s e c θ^{2} + {cot}^{2} θ + 2 c o s e c θ . cot θ}{1}$

$= 1 + {cot}^{2} θ + {cot}^{2} θ + 2 c o s e c θ . cot θ$
$= 1 + 2 {cot}^{2} θ + 2 c o s e c θ . cot θ$
$= R H S$

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