Question

What is the value of $\frac{\tan A-\sin A}{{\sin }^{3}A}$?

• A. $\frac{\sec A}{1-\cos A}$
• B. $\frac{\sec A}{1+{\cos }^{2}A}$
• C. $\frac{\sec A}{1+\cos A}$
• D. None of these

Answer 1

Answer: (A)

$\frac{\sec A}{1-\cos A}$

$\frac{\tan A-\sin A}{{\sin }^{3}A}=\frac{\frac{\sin A}{\cos A}-\sin A}{{\sin }^{3}A}=\frac{\frac{1-\cos A}{\cos A}}{{\sin }^{2}A}=\frac{\text{sec}A(1-\cos A)}{1-{\cos }^{2}A}$

$=\frac{\text{sec}A(1-\cos A)}{(1-\cos A)(1+\cos A)}$

$=\frac{\text{sec}A}{1+\cos A}$