$$\sqrt{\frac{\cot\theta+\cos\theta}{\cot\theta-\cos\theta}}$$ is equal to:
$$\sqrt{\frac{\cot\theta+\cos\theta}{\cot\theta-\cos\theta}}$$
=Â $$\sqrt{\frac{cos\theta(\frac{1}{sin\theta}+1)}{cos\theta(\frac{1}{sin\theta} - 1)}}$$
= $$\sqrt{\frac{1 + sin\theta}{1 - sin\theta} \times \frac{1 + sin\theta}{1 + sin\theta}}$$
= $$\sqrt{\frac{(1 + sin\theta)^2}{1 - sin^2\theta}}$$
= $$\frac{1 + sin\theta}{cos\theta}$$
= $$sec\theta + tan\theta$$
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