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Prove that : $\frac{\cos \theta}{1 - \tan \theta} + \frac{\sin \theta}{1 - \cos \theta} = \cos \theta + \sin \theta$.
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Solution: $LHS = \frac{\cos \theta}{1 - \frac{\sin \theta}{\cos \theta}} + \frac{\sin \theta}{1 - \frac{\cos \theta}{\sin \theta}} = \frac{\cos^2 \theta}{\cos \theta - \sin \theta} - \frac{\sin^2 \theta}{\cos \theta - \sin \theta} = \frac{(\cos \theta - \sin \theta)(\cos \theta + \sin \theta)}{\cos \theta - \sin \theta} = \cos \theta + \sin \theta = RHS$