What is the value of sin θ + cos θ, if θ satisfies the equation cot² θ - (√3 + 1) cot θ + √3 = 0; ?

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^∘ =

∴ The correct answer is .

Alternate Method

Given:

cot2 θ - ( + 1) cot θ + = 0

Calculation:

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

⇒ cot2 θ - ( + 1) cot θ + = cot² 30⁰ - ( + 1) cot 30⁰ +

⇒ cot2 θ - ( + 1) cot θ + = (3) - ( + 1) +

⇒ cot2 θ - ( + 1) cot θ + = 3 - (3 + ) +

⇒ cot2 θ - ( + 1) cot θ + = 3 - 3 - + = 0

Thus, the given condition satisfies

⇒ sin θ + cos θ = sin 30⁰ + cos 30⁰

⇒ sin θ + cos θ = 1/2 + /2 = ( + 1)/2

∴ The correct answer is .