List of top Mathematics Questions on Trigonometric Identities asked in UK Class X

Prove the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) for any right-angled triangle and use it to show that: \[ \frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1}{\sec \theta - \tan \theta}. \]

If \( \sin A + \cos A = \sqrt{2} \), find the value of \( \sin A \cos A \).