If (cos θ / (1 − sin θ)) + (cos θ / (1 + sin θ)) = 4 for an acute angle θ, then θ is equal to how many degrees?

Introduction / Context:
This trigonometric equation problem asks you to simplify a sum of two fractions involving sine and cosine and then solve for an acute angle θ. It tests your ability to manipulate trigonometric expressions and apply identities to reach a standard value that can be compared with known special angles.

Given Data / Assumptions:

• (cos θ / (1 − sin θ)) + (cos θ / (1 + sin θ)) = 4.
• 0° < θ < 90°, so θ is acute.
• Denominators 1 − sin θ and 1 + sin θ are non zero.

Concept / Approach:
Key concepts:

• Use a common denominator (1 − sin θ)(1 + sin θ).
• Apply the identity 1 − sin^2 θ = cos^2 θ.
• Simplify the resulting expression to involve only sec θ or cos θ.
• Equate the simplified form to 4 and solve for θ.

Step-by-Step Solution:
Let E = cos θ / (1 − sin θ) + cos θ / (1 + sin θ). Common denominator is (1 − sin θ)(1 + sin θ) = 1 − sin^2 θ = cos^2 θ. Numerator becomes cos θ(1 + sin θ) + cos θ(1 − sin θ). Factor cos θ: cos θ[(1 + sin θ) + (1 − sin θ)] = cos θ(2) = 2cos θ. Therefore E = (2cos θ) / cos^2 θ = 2 / cos θ = 2sec θ. Given that E = 4, we have 2sec θ = 4. So sec θ = 2, which implies cos θ = 1 / 2. For an acute angle, cos θ = 1 / 2 gives θ = 60°.

Verification / Alternative check:
Substitute θ = 60° back into the original expression. cos 60° = 1 / 2 and sin 60° = √3 / 2. Evaluate cos θ / (1 − sin θ) and cos θ / (1 + sin θ), then add them. After simplification, the result equals 4, confirming the correctness of θ = 60°.

Why Other Options Are Wrong:
Option b: 45° gives cos 45° = √2 / 2 and leads to a value that is not 4 when substituted.
Option c: 30° gives cos 30° = √3 / 2 and creates a different sum.
Option d and option e: 35° or 15° are not standard special angles that give sec θ = 2, and the equation would not hold for these values.

Common Pitfalls:
Errors often occur when finding the common denominator or expanding the numerators. Some students forget that (1 − sin θ)(1 + sin θ) equals cos^2 θ, or they try to solve the equation numerically instead of simplifying symbolically. Correct use of identities makes the problem straightforward.

Final Answer:
The acute angle θ satisfying the equation is 60°.