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Question
Prove the following.
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Solution
\[\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta}\]
\[ = \frac{1 + \sin\theta + 1 - \sin\theta}{\left( 1 - \sin\theta \right)\left( 1 + \sin\theta \right)}\]
\[ = \frac{2}{1 - \sin^2 \theta} \left[ \left( a - b \right)\left( a + b \right) = a^2 - b^2 \right]\]
\[ = \frac{2}{\cos^2 \theta} \left( \sin^2 \theta + \cos^2 \theta = 1 \right)\]
\[ = 2 \sec^2 \theta\]
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