Show that \(\frac{\sin \theta}{\sin \theta + \cos \theta} + \frac{\cos \theta}{\sin \theta - \cos \theta} \equiv \frac{1}{\sin^2 \theta - \cos^2 \theta}\).
Solution
Start by combining the fractions:
\(\frac{\sin \theta (\sin \theta - \cos \theta) + \cos \theta (\sin \theta + \cos \theta)}{(\sin \theta + \cos \theta)(\sin \theta - \cos \theta)}\)
Expand the numerator:
\(\sin^2 \theta - \sin \theta \cos \theta + \cos \theta \sin \theta + \cos^2 \theta\)
Simplify the numerator:
\(\sin^2 \theta + \cos^2 \theta\)
Since \(\sin^2 \theta + \cos^2 \theta = 1\), the expression becomes:
\(\frac{1}{\sin^2 \theta - \cos^2 \theta}\)
Log in to record attempts.