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Prove that : ...

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Prove that :
$\frac{s i n θ}{1 - c o s θ} = c o s e c θ + c o t θ$

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Solution

We have,
L.H.S. = $\frac{s i n θ}{1 - c o s θ}$

$\Rightarrow$ L.H.S. = $\frac{s i n θ}{(1 - c o s θ)} \times \frac{(1 + c o s θ)}{(1 + c o s θ)}$ ............ [Multiplying numerator and denomenator by $(1 - c o s θ)$ ]

$\Rightarrow$ L.H.S. = $\frac{s i n θ (1 + c o s θ)}{1 - c o s^{2} θ}$ $[∵ (1 + c o s θ) (1 - c o s θ) = 1 - c o s^{2} θ]$

$\Rightarrow$ L.H.S. = $\frac{s i n θ (1 + c o s θ)}{s i n^{2} θ}$ $[∵ 1 - c o s^{2} θ = s i n^{2} θ]$

$\Rightarrow$ L.H.S. = $\frac{1 + c o s θ}{s i n θ}$

$\Rightarrow$ L.H.S. = $\frac{1}{s i n t h e t a} + \frac{c o s θ}{s i n θ}$

$\Rightarrow$ L.H.S. = $c o s e c θ + c o t θ = R . H . S .$

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