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Prove that $(\sin \theta + \cos \theta) (\tan \theta + \cot \theta) = \sec \theta + \text{cosec } \theta$.
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LHS $= (\sin \theta + \cos \theta) (\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta})$
$= (\sin \theta + \cos \theta)(\frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta})$
$= \frac{(\sin \theta + \cos \theta).(1)}{\cos \theta \sin \theta}$
$= \sec \theta + \text{cosec } \theta = \text{RHS}$
$= (\sin \theta + \cos \theta)(\frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta})$
$= \frac{(\sin \theta + \cos \theta).(1)}{\cos \theta \sin \theta}$
$= \sec \theta + \text{cosec } \theta = \text{RHS}$