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Integration as Antiderivative

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Question

How do you prove $(\frac{c o s x}{1 + s i n x}) + (\frac{c o s x}{1 - s i n x}) = 2 s e c x$ ?

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Solution

LHS: $(\frac{c o s x}{1 + s i n x}) + (\frac{c o s x}{1 - s i n x}) = 2 s e c x$
$\frac{c o s x (1 - s i n x) + c o s x (1 + s i n x)}{1 - s i n^{2} x}$ --> common denominator
$= \frac{c o s x - s i n x c o s x + c o s x + s i n x c o s x}{c o s^{2} x}$
$= \frac{2 c o s x}{c o s^{2} x}$
$= \frac{2}{c o s x}$
$= 2 \cdot \frac{1}{c o s x}$
$= 2 s e c x$
$= R H S$

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