If cos x · cos y + sin x · sin y = −1, what is the value of cos x + cos y?

Introduction / Context:
This trigonometric identity question uses a well known formula for the cosine of a difference. You are given cos x · cos y + sin x · sin y and asked to compute cos x + cos y. Such questions test how well you know compound angle formulas and how to use them to deduce relationships between trigonometric expressions.

Given Data / Assumptions:

• cos x · cos y + sin x · sin y = −1.
• x and y are real angles.
• We need cos x + cos y.

Concept / Approach:
The identity cos(x − y) = cos x · cos y + sin x · sin y directly matches the left side. Once we know cos(x − y) = −1, we know that x − y must be an odd multiple of π (or 180°) and can then use the sum formula for cos x + cos y in terms of cos((x + y)/2) and cos((x − y)/2). That leads quickly to a product in which one factor is zero.

Step-by-Step Solution:
Recall the identity: cos(x − y) = cos x · cos y + sin x · sin y. Given cos x · cos y + sin x · sin y = −1, we have cos(x − y) = −1. The cosine of an angle equals −1 when the angle is an odd multiple of π (or 180°). Thus x − y = (2k + 1)π for some integer k. Now use the sum formula for cosines: cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2). We know x − y = (2k + 1)π, so (x − y)/2 = (2k + 1)π/2. Cosine of an odd multiple of π/2 is zero. Therefore cos((x − y)/2) = 0. So cos x + cos y = 2 cos((x + y)/2) · 0 = 0.
Verification / Alternative check:
Choose a simple pair of angles to test the result. Let y = 0 and x = π, then cos x · cos y + sin x · sin y = cos π·cos 0 + sin π·sin 0 = (−1)·1 + 0·0 = −1, which matches the given condition. Now cos x + cos y = cos π + cos 0 = −1 + 1 = 0, confirming the value.

Why Other Options Are Wrong:
Values such as ±2 would require both cosines to be ±1 simultaneously, which is impossible if cos(x − y) is −1. A value 1 or −1 would correspond to partial cancellation rather than total cancellation. The identity and sample check both show that the sum must be exactly zero.

Common Pitfalls:
A frequent mistake is to misremember the identity and use cos(x + y) instead of cos(x − y). Another issue is forgetting to use the product-to-sum formula for cos x + cos y, which makes it harder to see that a factor becomes zero. Keeping a clear list of standard identities helps avoid such confusion.

Final Answer:
The value of cos x + cos y is 0.