Section A: Board level Questions
- If $\sin \theta + \cos \theta = \sqrt{2}\cos\theta$, prove that $\cos \theta - \sin \theta = \sqrt{2}\sin\theta$.
- Evaluate the following expression: $\frac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$.
- Prove the identity: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \csc \theta$.
- If $\sec \theta + \tan \theta = p$, prove that $\sin \theta = \frac{p^2 - 1}{p^2 + 1}$.
- Prove: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$.
