If the trigonometric equation 2 cos θ = 2 − sin θ holds for an angle θ in a range where both sine and cosine are defined, what are the possible values of cos θ that satisfy this relationship?

Introduction / Context:
This question tests algebraic manipulation of trigonometric equations and the use of the basic identity sin^2θ + cos^2θ = 1. You are asked to solve the given equation for cos θ and determine all possible values that satisfy both the equation and the fundamental trigonometric identity.

Given Data / Assumptions:

• The equation is 2 cos θ = 2 − sin θ.
• The angle θ is such that sin θ and cos θ are both defined.
• We must find the possible values of cos θ that satisfy the equation and the identity sin^2θ + cos^2θ = 1.
• We assume standard real valued trigonometric functions.

Concept / Approach:
We treat sine and cosine as algebraic variables linked by the identity sin^2θ + cos^2θ = 1. From the equation 2 cos θ = 2 − sin θ, we express cos θ in terms of sin θ, then substitute this into the identity. This produces a quadratic equation in sin θ, which we solve to find possible sine values. For each sine value, we then use the original equation to find the corresponding cos θ values.

Step-by-Step Solution:
Start from 2 cos θ = 2 − sin θ and divide by 2: cos θ = 1 − (sin θ)/2.Let s = sin θ. Then cos θ = 1 − s/2. Use the identity sin^2θ + cos^2θ = 1, which becomes s^2 + (1 − s/2)^2 = 1.Expand (1 − s/2)^2: this is 1 − s + s^2/4. So the equation is s^2 + 1 − s + s^2/4 = 1.Combine like terms: s^2 + s^2/4 = (5/4)s^2. So (5/4)s^2 − s + 1 = 1, which simplifies to (5/4)s^2 − s = 0.Factor: s((5/4)s − 1) = 0, giving s = 0 or s = 4/5. For s = 0, 2 cos θ = 2, so cos θ = 1. For s = 4/5, 2 cos θ = 2 − 4/5 = 6/5, so cos θ = 3/5.

Verification / Alternative check:
Check each candidate pair (sin θ, cos θ) against the identity. For sin θ = 0 and cos θ = 1, we have sin^2θ + cos^2θ = 0 + 1 = 1, which is correct. For sin θ = 4/5 and cos θ = 3/5, sin^2θ + cos^2θ = (16/25) + (9/25) = 25/25 = 1, also correct. Substituting back into the original equation 2 cos θ = 2 − sin θ confirms that both cos θ = 1 and cos θ = 3/5 satisfy the equation.

Why Other Options Are Wrong:

• Options involving −1 or −1/2 suggest sine values that either violate the equation 2 cos θ = 2 − sin θ or the identity sin^2θ + cos^2θ = 1.
• Option e, 3/5 only, ignores the valid solution cos θ = 1, which is obtained when sin θ = 0.
• Combinations like 1 or −1/2 and −1 or 3/5 are not supported by the algebraic derivation.

Common Pitfalls:

• Squaring the equation incorrectly or forgetting to apply the identity sin^2θ + cos^2θ = 1.
• Dropping one of the solutions from the quadratic equation in sin θ.
• Failing to check the obtained sine and cosine values against both the identity and the original equation.

Final Answer:
1 or 3/5