Prove the identity \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \equiv \frac{\tan \theta - 1}{\tan \theta + 1}\).
Solution
Start with \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta}\).
Divide the numerator and the denominator by \(\cos \theta\):
\(\frac{\frac{\sin \theta}{\cos \theta} - 1}{\frac{\sin \theta}{\cos \theta} + 1}\).
Let \(t = \tan \theta = \frac{\sin \theta}{\cos \theta}\), then:
\(\frac{t - 1}{t + 1}\).
This matches the right-hand side of the identity.
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