(sec A + tan A) (1 – sin A) equals:

To solve the given expression, let's break it down step by step:

1. Start with the expression: (sec A + tan A) (1 – sin A)

2. Rewrite sec A as 1/cos A: (1/cos A + tan A) (1 – sin A)

3. Simplify the expression by finding a common denominator for the terms in the numerator: (1 + cos A sin A / cos A) (1 – sin A)

4. Combine the terms in the numerator: (1 + cos A sin A) (1 – sin A)

5. Expand the expression using the distributive property: 1 - sin A + cos A sin A - sin^2 A

6. Simplify further by combining like terms: 1 - sin A + cos A sin A - sin^2 A

7. Rearrange the terms: 1 - sin^2 A + cos A sin A - sin A

8. Use the identity sin^2 A + cos^2 A = 1 to replace sin^2 A with 1 - cos^2 A: 1 - (1 - cos^2 A) + cos A sin A - sin A

9. Simplify the expression: 1 - 1 + cos^2 A + cos A sin A - sin A

10. Combine like terms: cos^2 A + cos A sin A - sin A

Therefore, the expression (sec A + tan A) (1 – sin A) simplifies to cos^2 A + cos A sin A - sin A.