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Prove the following trigonometric identity :
$\frac{\cos \theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{\cos \theta} = 2 \sec \theta$
$\frac{\cos \theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{\cos \theta} = 2 \sec \theta$
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Solution: LHS: $\frac{\cos^2 \theta + (1 + \sin \theta)^2}{(1 + \sin \theta) \cos \theta}$ [1 mark]
$= \frac{\cos^2 \theta + 1 + \sin^2 \theta + 2 \sin \theta}{(1 + \sin \theta) \cos \theta}$ [1/2 mark]
$= \frac{2 (1 + \sin \theta)}{(1 + \sin \theta) \cos \theta}$ [1 mark]
$= 2 \sec \theta = RHS$ [1/2 mark]
$= \frac{\cos^2 \theta + 1 + \sin^2 \theta + 2 \sin \theta}{(1 + \sin \theta) \cos \theta}$ [1/2 mark]
$= \frac{2 (1 + \sin \theta)}{(1 + \sin \theta) \cos \theta}$ [1 mark]
$= 2 \sec \theta = RHS$ [1/2 mark]