a) If \( \sin A=\frac{4}{5} \), find the value of \( \cot A+\tan A \). b) If \( \sin A=\frac{3}{8} \), find the value of \( \operatorname{cosec} A-\sec A \).

To find \( \cot A + \tan A \) when \( \sin A = \frac{4}{5} \): 1. Use the identity \( \cos^2 A + \sin^2 A = 1 \) to find \( \cos A \): \[ \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \] So, \( \cos A = \frac{3}{5} \). 2. Calculate \( \tan A \) and \( \cot A \): \[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \] \[ \cot A = \frac{1}{\tan A} = \frac{3}{4} \] 3. Combine the two: \[ \cot A + \tan A = \frac{3}{4} + \frac{4}{3} \] To add, find a common denominator (which is 12): \[ \cot A + \tan A = \frac{9}{12} + \frac{16}{12} = \frac{25}{12} \] Thus, \( \cot A + \tan A = \frac{25}{12} \). Now for \( \operatorname{cosec} A - \sec A \) when \( \sin A = \frac{3}{8} \): 1. Calculate \( \csc A \) and \( \sec A \): \[ \csc A = \frac{1}{\sin A} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \] To find \( \cos A \), use the Pythagorean identity: \[ \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{3}{8}\right)^2 = 1 - \frac{9}{64} = \frac{55}{64} \] So, \( \cos A = \frac{\sqrt{55}}{8} \). 2. Now calculate \( \sec A \): \[ \sec A = \frac{1}{\cos A} = \frac{8}{\sqrt{55}} \] 3. Combine \( \csc A - \sec A \): \[ \csc A - \sec A = \frac{8}{3} - \frac{8}{\sqrt{55}} \] To get a common denominator: \[ = \frac{8\sqrt{55}}{3\sqrt{55}} - \frac{24}{3\sqrt{55}} = \frac{8\sqrt{55} - 24}{3\sqrt{55}} \] Thus, \( \operatorname{cosec} A - \sec A = \frac{8\sqrt{55} - 24}{3\sqrt{55}} \).