For an acute angle A, suppose sec A + tan A = x. Using standard identities, determine which expression in terms of sine and cosine of A is equal to x.

Introduction / Context:
This trigonometry question examines your ability to manipulate expressions involving secant and tangent and rewrite them in terms of sine and cosine. You are told that sec A + tan A equals x for an acute angle A and asked to identify another expression that is equal to x. This requires both algebraic manipulation and knowledge of trigonometric identities, which are important skills in higher level mathematics and aptitude examinations.

Given Data / Assumptions:

• A is an acute angle.
• sec A + tan A = x.
• sec A = 1/cos A and tan A = sin A/cos A.
• sine and cosine satisfy sin^2 A + cos^2 A = 1.
• We must find an expression in terms of sin A and cos A that is equal to x.

Concept / Approach:
First rewrite sec A and tan A using sine and cosine. This gives a single fraction with cos A as the denominator. Next, rationalise this expression by multiplying the numerator and denominator by a suitable conjugate in order to transform the denominator into a simple expression in sin A. After simplification, we look for a match with one of the given options. Using the identity sin^2 A + cos^2 A = 1 is crucial for reducing the expression to its most useful form.

Step-by-Step Solution:
Step 1: Express sec A and tan A in terms of sine and cosine: sec A = 1/cos A and tan A = sin A/cos A.Step 2: Add them: sec A + tan A = 1/cos A + sin A/cos A = (1 + sin A)/cos A.Step 3: So x = (1 + sin A)/cos A.Step 4: Multiply numerator and denominator by (1 − sin A) to transform the expression: x = [(1 + sin A)(1 − sin A)] / [cos A(1 − sin A)].Step 5: Use the identity (1 + sin A)(1 − sin A) = 1 − sin^2 A.Step 6: Since 1 − sin^2 A = cos^2 A, the numerator becomes cos^2 A.Step 7: Therefore x = cos^2 A / [cos A(1 − sin A)] = cos A / (1 − sin A).

Verification / Alternative check:
Pick a simple acute angle, such as A = 30°. Then sec 30° = 1/cos 30° = 2/√3 and tan 30° = 1/√3. Their sum is (2/√3 + 1/√3) = 3/√3 = √3. Now evaluate cos A/(1 − sin A): cos 30° = √3/2 and sin 30° = 1/2, so cos A/(1 − sin A) = (√3/2)/(1 − 1/2) = (√3/2)/(1/2) = √3. The two results match, confirming that sec A + tan A equals cos A/(1 − sin A).

Why Other Options Are Wrong:
The expression cos A/(1 + sin A) is the reciprocal of cos A/(1 − sin A) after rationalisation and corresponds more closely to sec A − tan A. The square root expressions √(cos A/(1 − sin A)) and √(cos A/(1 + sin A)) introduce a square root that is not present in the original x and therefore cannot be equal. The option (1 + sin A)/cos A is the unreduced form we obtained before rationalisation, but among the listed choices, the fully simplified identity that matches standard forms is cos A/(1 − sin A).

Common Pitfalls:
Students sometimes stop at the intermediate form (1 + sin A)/cos A and do not recognise that rationalising further can produce a more useful equivalent expression. Others may incorrectly multiply numerator and denominator or misapply the identity 1 − sin^2 A = cos^2 A. Carefully applying each algebraic step and checking with a specific example angle helps solidify the understanding of these manipulations.

Final Answer:
When sec A + tan A = x, an equivalent expression is cos A / (1 − sin A).