If sin θ + cos θ = √3 cos θ, then find the value of cot θ.

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Given: sin θ + cos θ = √3 cos θ

Step 1: Rearrange the equation:
 sin θ = √3 cos θ - cos θ
 sin θ = cos θ (√3 - 1)

Step 2: Divide both sides by cos θ (assuming cos θ ≠ 0):
 (sin θ) / (cos θ) = √3 - 1

Step 3: Recall that (sin θ) / (cos θ) = tan θ:
 tan θ = √3 - 1

Step 4: We need cot θ, which is the reciprocal of tan θ:
 cot θ = 1 / tan θ = 1 / (√3 - 1)

Step 5: Rationalize the denominator:
 cot θ = (1 / (√3 - 1)) × ((√3 + 1) / (√3 + 1)) = (√3 + 1) / ((√3)^2 - 1^2) = (√3 + 1) / (3 - 1) = (√3 + 1) / 2

Therefore, cot θ = (√3 + 1) / 2.

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