If A is an acute angle, simplify the trigonometric expression cos^4 A − sin^4 A and express your answer in terms of a standard trigonometric function of 2A.

Introduction / Context:
This question checks familiarity with basic trigonometric identities and algebraic factorisation. The expression cos^4 A − sin^4 A resembles the difference of two fourth powers, which is naturally treated as a difference of squares. The goal is to connect the result to a well known double angle identity involving cos 2A. Such transformations are very common in trigonometry sections of aptitude and entrance exams.

Given Data / Assumptions:

• A is an acute angle, though the identity will hold for any real angle.
• The expression to simplify is cos^4 A − sin^4 A.
• We may use standard identities such as cos^2 A + sin^2 A = 1 and cos 2A = cos^2 A − sin^2 A.
• The final answer must be expressed using a standard trigonometric function, preferably cos 2A.

Concept / Approach:
The main idea is to factor cos^4 A − sin^4 A as a difference of squares. We treat cos^4 A as (cos^2 A)^2 and sin^4 A as (sin^2 A)^2. The difference of squares formula a^2 − b^2 = (a − b)(a + b) can then be applied. Once we have a product of two simpler expressions, we can replace cos^2 A − sin^2 A by cos 2A and cos^2 A + sin^2 A by 1, leading to a very compact result.

Step-by-Step Solution:
Step 1: Rewrite cos^4 A as (cos^2 A)^2 and sin^4 A as (sin^2 A)^2.Step 2: Use the difference of squares identity: a^2 − b^2 = (a − b)(a + b).Step 3: Therefore cos^4 A − sin^4 A = (cos^2 A − sin^2 A)(cos^2 A + sin^2 A).Step 4: Use the Pythagorean identity cos^2 A + sin^2 A = 1.Step 5: This simplifies the expression to cos^2 A − sin^2 A.Step 6: Recognise that cos 2A = cos^2 A − sin^2 A.Step 7: Hence cos^4 A − sin^4 A = cos 2A.

Verification / Alternative check:
Choose an easy angle such as A = 30°. Then cos A = √3/2 and sin A = 1/2. Compute cos^4 30° − sin^4 30°. We have cos^2 30° = 3/4 and sin^2 30° = 1/4, so cos^4 30° = 9/16 and sin^4 30° = 1/16. The difference is 8/16 = 1/2. Now compute cos 2A = cos 60° = 1/2. Both results match, confirming our identity.

Why Other Options Are Wrong:
The option 0 would be correct only if cos^4 A and sin^4 A were equal for all A, which they are not. The option 2 cos 2A is twice the correct value and arises if one mistakenly multiplies by 2 when converting to double angle form. The option 1 ignores the dependence on A entirely. The option sin 2A is unrelated to cos^4 A − sin^4 A under the standard identities. Only cos 2A matches the correct algebraic simplification.

Common Pitfalls:
A frequent error is to try to convert cos^4 A and sin^4 A directly into double angle forms using power reduction formulas, which is unnecessarily long here. Some students also forget that cos^2 A + sin^2 A = 1 and therefore miss the simple factor that collapses the expression. Recognising the structure of a difference of squares and then applying standard identities is the most efficient route.

Final Answer:
The expression cos^4 A − sin^4 A simplifies to cos 2A.