The value of $1+\sqrt{\frac{\cot \theta+\cos \theta}{\cot \theta-\cos \theta}}$, if $0^{\circ}<\theta<90^{\circ}$, is equal to:

$1+\sec \theta+\tan \theta$

1 + $\sqrt{\frac{cotθ+cosθ}{cotθ-cosθ}}$

=1 + $\sqrt{\frac{1+sinθ}{1-sinθ}} × \sqrt{\frac{1+sinθ}{1+sinθ}}$

= 1 + $\sqrt{\frac{(1+sinθ)²}{1²-sin²θ}} $

= 1 + $\sqrt{\frac{(1+sinθ)²}{cos²θ}} $

= 1 + secθ + tanθ