Question

Evaluate \( (\sec A + \tan A)(1 - \sin A) = \ ? \)

• A. \( \sin A \)
• B. \( \cos A \)
• C. \( \csc A \)
• D. \( \sec A \)

Hint:

Always look to apply standard identities such as: - \( \sec A = \frac{1}{\cos A} \), \( \tan A = \frac{\sin A}{\cos A} \) - \( (a + b)(a - b) = a^2 - b^2 \) - \( \sin^2 A + \cos^2 A = 1 \)

Answer 1

The Correct Option is B

Step 1: Use identities to simplify. We use: \[ \sec A = \frac{1}{\cos A}, \quad \tan A = \frac{\sin A}{\cos A} \] So, \[ (\sec A + \tan A)(1 - \sin A) = \left( \frac{1}{\cos A} + \frac{\sin A}{\cos A} \right)(1 - \sin A) \] \[ = \frac{1 + \sin A}{\cos A} \cdot (1 - \sin A) \] Step 2: Apply identity \( (a + b)(a - b) = a^2 - b^2 \): \[ = \frac{(1 + \sin A)(1 - \sin A)}{\cos A} = \frac{1 - \sin^2 A}{\cos A} \] Step 3: Use Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \): \[ 1 - \sin^2 A = \cos^2 A \] So, \[ \frac{\cos^2 A}{\cos A} = \cos A \]