If sec θ + tan θ = 2 + √5, then the value of sin θ + cos θ is :

given sec θ + tan θ = 2 + √5-------(1)

we know that, 

$$sec^{2}θ-tan^{2}θ=1$$

$$(sec θ + tan θ).(sec θ - tan θ)=1$$

(2 + √5).(sec θ - tan θ)=1

(sec θ - tan θ)=√5-2 ------(2)

on solving (1)&(2)

secθ=√5 $$\Rightarrow$$ cosθ=1/√5

tan θ=2 $$\Rightarrow$$ sin θ/cosθ=2 $$\Rightarrow$$ sinθ=2.cosθ $$\Rightarrow$$ sinθ =2/√5

$$\Rightarrow$$ sin θ + cos θ = 1/√5 + 2/√5

$$\Rightarrow$$ sin θ + cos θ =  3/√5

so the answer is option A.