If sin θ + cos θ = 1 for an angle θ (in degrees), then what is the value of the product sin θ · cos θ?

Introduction / Context:
This trigonometric simplification question tests the use of the Pythagorean identity sin^2 θ + cos^2 θ = 1, together with basic algebra. It is a classic type of problem used to check understanding of relationships between sine and cosine and the effect of squaring an equation involving them.

Given Data / Assumptions:

sin θ + cos θ = 1.
θ is an angle measured in degrees or radians, with standard trigonometric definitions.
We need to find sin θ · cos θ.

Concept / Approach:
The idea is to square both sides of the equation sin θ + cos θ = 1. When we square the left side, we obtain sin^2 θ + cos^2 θ plus twice the product sin θ cos θ. Using the identity sin^2 θ + cos^2 θ = 1 allows us to solve for sin θ cos θ directly. This method is very efficient and requires no explicit angle values.

Step-by-Step Solution:
Starting from sin θ + cos θ = 1, square both sides.Left side: (sin θ + cos θ)^2 = sin^2 θ + 2 sin θ cos θ + cos^2 θ.Right side: 1^2 = 1.Use the identity sin^2 θ + cos^2 θ = 1, so the left side becomes 1 + 2 sin θ cos θ.Set this equal to 1: 1 + 2 sin θ cos θ = 1.Subtract 1 from both sides: 2 sin θ cos θ = 0.Therefore sin θ cos θ = 0.

Verification / Alternative check:
If sin θ cos θ = 0, then at least one of sin θ or cos θ must be zero. Consider θ = 0°, then sin θ = 0 and cos θ = 1, and sin θ + cos θ = 0 + 1 = 1, which satisfies the given condition. Similarly, θ = 90° gives sin θ = 1 and cos θ = 0, and their sum is again 1. In both cases, the product sin θ cos θ equals 0, confirming our algebraic result.

Why Other Options Are Wrong:
If sin θ cos θ were 1, 1/2, −1/2, or −1, then the squared equation would become 1 + 2 sin θ cos θ = 1, which would not hold. For example, if sin θ cos θ = 1/2, the left side becomes 1 + 1 = 2, which is not equal to 1. Hence all other options are inconsistent with the relation sin θ + cos θ = 1.

Common Pitfalls:
Students sometimes attempt to guess an angle without using algebra, or they mistakenly square each term separately without considering the cross term 2 sin θ cos θ. Remember that (a + b)^2 expands to a^2 + 2ab + b^2. Also, forgetting the fundamental identity sin^2 θ + cos^2 θ = 1 can prevent reaching the correct conclusion quickly.

Final Answer:
The product sin θ · cos θ is 0.