1. If \(\angle A + \angle B = 90^\circ\), then prove that \[ 1 + \cfrac{\tan A}{\tan B} = \tan^2 A \sec^2 B \]

2. If \(\angle P + \angle Q = 90^\circ\), then **prove** that: \[ \sqrt{\frac{\sin P}{\cos Q} - \sin P \cos Q} = \cos P \]

3. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.

4. If ∠A + ∠B = 90°, then prove that \[ 1 + \frac{\tan A}{\tan B} = \sec^2 A \]

5. In triangle ABC, \(\angle BAC\) is a right angle. If CD is the median, then prove that \[ BC^2 = CD^2 + 3AD^2 \]