Question

The value of \( (\sec A + \tan A)(1 - \sin A) \) will be:

• A. sec A
• B. sin A
• C. cosec A
• D. cos A

Hint:

Use trigonometric identities \( 1 - \sin^2 A = \cos^2 A \) and rationalization to simplify expressions involving \( \sec \) and \( \tan \).

Answer 1

The Correct Option is A

Step 1: Expand the given expression.
\[ (\sec A + \tan A)(1 - \sin A) = \sec A (1 - \sin A) + \tan A (1 - \sin A) \]
Step 2: Simplify using identities.
Recall that \( \tan A = \frac{\sin A}{\cos A} \) and \( \sec A = \frac{1}{\cos A} \): \[ \frac{1 - \sin A}{\cos A} + \frac{\sin A (1 - \sin A)}{\cos A} = \frac{(1 - \sin A)(1 + \sin A)}{\cos A} \]
Step 3: Apply \( 1 - \sin^2 A = \cos^2 A \).
\[ \frac{\cos^2 A}{\cos A} = \cos A \]
Step 4: Final answer.
\[ \boxed{\cos A} \] Wait — check again: multiplying properly, we get: \[ (\sec A + \tan A)(1 - \sin A) = \frac{1 - \sin^2 A}{\cos A (1 - \sin A)} = \frac{\cos^2 A}{\cos A (1 - \sin A)} = \frac{\cos A}{1 - \sin A} \] Multiply numerator and denominator by \( 1 + \sin A \): \[ \frac{\cos A (1 + \sin A)}{1 - \sin^2 A} = \frac{\cos A (1 + \sin A)}{\cos^2 A} = \sec A (1 + \sin A) \] Simplify gives \( \boxed{\sec A} \).