What is the simplified value of {sin (90 – x) cos [π – (x – y)]} + {cos (90 – x) sin [π – (y – x)]} in trigonometry?

Introduction / Context:
This trigonometric expression involves complementary angles (90 degrees minus x) and supplementary angles (π minus something). It is constructed so that the sum can be converted into a simple function of y using standard identities and the sine of a sum formula.

Given Data / Assumptions:

• Expression E = sin(90 - x) cos[π - (x - y)] + cos(90 - x) sin[π - (y - x)].
• x and y are real angles.
• We are required to simplify E.

Concept / Approach:
We use standard identities: sin(90 - x) = cos x, cos(90 - x) = sin x, cos(π - u) = -cos u, sin(π - u) = sin u. After simplifying each term, the expression takes the form sin A cos B + cos A sin B, which is exactly the pattern for sin(A + B). Combining everything yields a compact expression involving cos y and a negative sign.

Step-by-Step Solution:
First, use sin(90 - x) = cos x. So the first factor in the first term becomes cos x. Next, rewrite cos[π - (x - y)]. Using cos(π - u) = -cos u, we have cos[π - (x - y)] = -cos(x - y). Therefore the first term is sin(90 - x) cos[π - (x - y)] = cos x * ( -cos(x - y) ) = -cos x cos(x - y). Now consider cos(90 - x) = sin x. For sin[π - (y - x)], use sin(π - u) = sin u, so sin[π - (y - x)] = sin(y - x). Thus the second term is cos(90 - x) sin[π - (y - x)] = sin x * sin(y - x). Note that sin(y - x) = -sin(x - y). Hence the second term equals sin x * ( -sin(x - y) ) = -sin x sin(x - y). So E = -cos x cos(x - y) - sin x sin(x - y). Factor out -1: E = -[cos x cos(x - y) + sin x sin(x - y)]. Use the identity cos(A - B) = cos A cos B + sin A sin B with A = x and B = x - y. Then cos(x - (x - y)) = cos y = cos x cos(x - y) + sin x sin(x - y). Therefore the bracket equals cos y, so E = -cos y.

Verification / Alternative check:
Choose specific values such as x = 40 degrees and y = 20 degrees. Compute the original expression numerically and compare with -cos y. Both values match within rounding error, confirming that E simplifies to -cos y.

Why Other Options Are Wrong:
cos x: This would arise if the final identity were misapplied, using the wrong angles in the cosine of a difference formula.
-sin y or sin y: These appear if one incorrectly relates the expression to a sine pattern instead of the cosine pattern.
tan y: This comes from confusing the sine of a sum formula with a tangent identity and does not match the structure of the expression.

Common Pitfalls:
Typical mistakes include sign errors when using cos(π - u) = -cos u, or forgetting that sin(y - x) equals -sin(x - y). Another pitfall is failing to recognize the pattern sin A cos B + cos A sin B hidden inside the expression. Writing each identity clearly and simplifying step by step avoids these issues.

Final Answer:
The simplified value of the expression is -cos y.