What is the value of sin θ + cos θ, if θ satisfies the equation cot² θ - (√3 + 1) cot θ + √3 = 0; \(0<\theta<\frac{\pi}{4} \)?

Shortcut Trick Here, 0 < θ < 45^∘

⇒ So possible value of θ = 30^∘

Putting θ = 30^∘ in the equation we get,

⇒ cot² 30^∘ - (√3 + 1) cot 30^∘ + √3

⇒ 3 - 3 - √3 + √3 = 0

⇒ So, sin 30^∘ + cos 30^∘ = \(\frac{\sqrt3 + 1}{2}\)

∴ The correct answer is \(\frac{\sqrt3 + 1}{2}\).

Alternate Method

Given:

cot2 θ - (\(√{3}\) + 1) cot θ + \(√{3}\) = 0

\(0<θ<\frac{\pi}{4} \)

Calculation:

Since θ lies between 0⁰ and 45⁰, putting θ = 30⁰

⇒ cot2 θ - (\(√{3}\) + 1) cot θ + \(√{3}\) = cot² 30⁰ - (\(√{3}\) + 1) cot 30⁰ + \(√{3}\)

⇒ cot2 θ - (\(√{3}\) + 1) cot θ + \(√{3}\) = (3) - (\(√{3}\) + 1)\(√{3}\) + \(√{3}\)

⇒ cot2 θ - (\(√{3}\) + 1) cot θ + \(√{3}\) = 3 - (3 + \(√{3}\)) + \(√{3}\)

⇒ cot2 θ - (\(√{3}\) + 1) cot θ + \(√{3}\) = 3 - 3 - \(√{3}\) + \(√{3}\) = 0

Thus, the given condition satisfies

⇒ sin θ + cos θ = sin 30⁰ + cos 30⁰

⇒ sin θ + cos θ = 1/2 + \(√{3}\)/2 = (\(√{3}\) + 1)/2

∴ The correct answer is \(\frac{√3\ +\ 1}{2}\).