Given that cos z = L , whrere z is an acute angle, find an expression for \(\frac{\cot z - \csc z}{\sec z + \tan z}\)
A. \(\frac{1 - L}{1 + L}\) B. \(\frac{L^2 \sqrt{3}}{1 + L}\) C. \(\frac{1 + L^3}{L^2}\) D. \(\frac{L(L - 1)}{1 - L + 1 \sqrt{1 - L^2}}\)
Correct Answer: D
Explanation
Given Cos z = L, z is an acute angle \(\frac{\text{cot z - cosec z}}{\text{sec z + tan z}}\) = cos z = \(\frac{\text{cos z}}{\text{sin z}}\) cosec z = \(\frac{1}{\text{sin z}}\) cot z - cosec z = \(\frac{\text{cos z}}{\text{sin z}}\) - \(\frac{1}{\text{sin z}}\) cot z - cosec z = \(\frac{L - 1}{\text{sin z}}\) sec z = \(\frac{1}{\text{cos z}}\) tan z = \(\frac{\text{sin z}}{\text{cos z}}\) sec z = \(\frac{1}{\text{cos z}}\) + \(\frac{\text{sin z}}{\text{cos z}}\) = \(\frac{1}{l}\) + \(\frac{\text{sin z}}{L}\) the original eqn. becomes \(\frac{\text{cot z - cosec z}}{\text{sec z + tan z}}\) = \(\frac{L - \frac{1}{\text{sin z}}}{1 + sin \frac{z}{L}}\) = \(\frac{L(L - 1)}{\text{sin z}(1 + \text{sin z})}\) = \(\frac{L(L - 1)}{\text{sin z} + 1 - cos^2 z}\) = sin z + 1 = 1 + \(\sqrt{1 - L^2}\) = \(\frac{L(L - 1)}{1 - L + 1 \sqrt{1 - L^2}}\)
