If sinθ + cosθ = √3, then find the value of tanθ + cotθ?

Given: 

sinθ + cosθ = √3

We need to find the value of tanθ + cotθ.

Concept Used:

Using trigonometric identities:

• (sinθ + cosθ)² = sin²θ + cos²θ + 2sinθcosθ

• sin²θ + cos²θ = 1

Calculation:

Step 1: Expand (sinθ + cosθ)²:
√3² = sin²θ + cos²θ + 2sinθ cosθ
⇒ 3 = 1 + 2sinθ cosθ
⇒ 2sinθ cosθ = 2
⇒ sinθ cosθ = 1

Step 2: Use the identity tanθ + cotθ:
tanθ + cotθ = (sinθ/cosθ) + (cosθ/sinθ)
⇒ tanθ + cotθ = (sin²θ + cos²θ) / (sinθcosθ)
⇒ tanθ + cotθ = 1 / 1
⇒ tanθ + cotθ = 1

Conclusion:

∴ The value of tanθ + cotθ is 1.

The correct answer is Option 3.