Using trigonometric sum formulas, find the exact value of the expression sin 46° cos 44° + cos 46° sin 44°.

Introduction / Context:
This question is designed to test your knowledge of trigonometric angle sum identities, especially the sine of a sum formula. Instead of directly calculating each term numerically, you can recognise the structure sin A cos B + cos A sin B and simplify it using the identity for sin(A + B). Recognising this pattern makes the problem very quick in an exam setting.

Given Data / Assumptions:

• The expression is sin 46° cos 44° + cos 46° sin 44°.
• All angles are in degrees.
• We are asked to find the exact value of this expression.

Concept / Approach:
The key identity is the sine addition formula: sin(A + B) = sin A cos B + cos A sin B. The given expression exactly matches this pattern if we choose A = 46° and B = 44°. Therefore, instead of treating the two products separately, we combine them into a single sine term. After that, we just evaluate sin of a simple angle (A + B), which here becomes sin 90°, a standard value.

Step-by-Step Solution:
Start with the identity: sin(A + B) = sin A cos B + cos A sin B. In the given expression, identify A = 46° and B = 44°. Then sin 46° cos 44° + cos 46° sin 44° = sin(46° + 44°). Compute the sum of angles: 46° + 44° = 90°. So the expression simplifies to sin 90°. We know sin 90° = 1. Therefore, the exact value of the original expression is 1.

Verification / Alternative check:
As an approximate check, we can think about approximate values. sin 46° and cos 44° are both close to sin 45° and cos 45°, which are about 0.707. Their product is roughly 0.5. The same is true for cos 46° and sin 44°. Adding these two approximate halves gives us a value close to 1. This numerical reasoning supports the exact value obtained from the identity.

Why Other Options Are Wrong:
sin 2° and cos 2° correspond to sin(A - B) and cos(A - B) type expressions, not sin(A + B). 2 and 0 are not valid values of sine for angles in degrees, except in trivial or impossible cases for standard trigonometric values. Sine of an angle is always between -1 and 1, so 2 is impossible, and 0 would correspond to angles like 0° or 180°, not 90°.

Common Pitfalls:
Students sometimes confuse sin(A + B) with cos(A + B) or with sin(A - B), leading to incorrect use of formulas. Another mistake is to try to approximate each term individually on a calculator instead of spotting the identity, which is slower and more error prone in exams. Always scan for standard patterns such as sin A cos B + cos A sin B when you see this kind of sum.

Final Answer:
The expression evaluates exactly to 1.