Question

\[\frac{1}{1 - \sin A}\] is equal to ______.

Options

• 1 + sin A

• sec² A (1 + sin A)

• sec² A

• sec² A + sin A

Answer 1

\[\frac{1}{1 - \sin A}\] is equal to sec² A (1 + sin A).

Explanation:

Solving,

\[\Rightarrow \frac{1}{1 - \sin A}\]

Multiplying numerator and denominator by 1 + sin A, we get :

\[\Rightarrow \frac{1}{1 - \sin A} \times \frac{1 + \sin A}{1 + \sin A}\]

\[\Rightarrow \frac{1 + \sin A}{1 - \sin^2 A}\]

\[\Rightarrow \frac{1 + \sin A}{\cos^2 A}\]

\[\Rightarrow \frac{1}{\cos^2 A} + \frac{\sin A}{\cos^2 A}\]

\[\Rightarrow \frac{1}{\cos^2 A} + \frac{1}{\cos^2 A} \cdot \sin A\]

⇒ sec²A + sec²A. sin A

⇒ sec²A(1 + sin A)