Differentiation - Derivatives of Reciprocal Trigonometric Functions

Mrs Koh wants to hang a decoration on the vertical wall above her bookshelf. She needs a ladder to climb up.

The rectangle \(ABCD\) is the side-view of the bookshelf and \(HK\) is the side-view of the ladder where \(AB = 24\text{ cm}\) and \(BC = 192\text{ cm}\) (see Figure 1). The ladder touches the wall at \(H\), the edge of the top of the bookshelf at \(B\) and the floor at \(K\).

1. Given that \(\angle HKD = \theta\), show that the length, \(L\text{ cm}\) of the ladder is given by \[L = \frac{24}{\cos \theta} + \frac{192}{\sin \theta}.\][2]
2. Use differentiation to find the exact value of the shortest length of the ladder as \(\theta\) varies.
   [You do not need to verify that this length of the ladder is the shortest.][5]