Answer
$$\frac{1}{\sin\theta\cos\theta}$$
$$or$$
$$\csc\theta\sec\theta$$
Work Step by Step
$$\cot\theta+\frac{1}{\cot\theta}$$
Multiply $\cot\theta$ by $\frac{\cot\theta}{\cot\theta}$ to get common denominators:
$$\frac{\cot^{2}\theta}{\cot\theta}+\frac{1}{\cot\theta}$$
$$=\frac{\cot^{2}\theta+1}{\cot\theta}$$
$\cot^{2}\theta+1=\csc^{2}\theta$ because of the Pythagorean Identities
$$=\frac{\csc^{2}\theta}{\cot\theta}$$
Rewrite using Reciprocal Identities:
$$=\frac{1}{\sin^{2}\theta}\div\frac{\cos\theta}{\sin\theta}$$
$$=\frac{1}{\sin^{2}\theta}\times\frac{\sin\theta}{\cos\theta}$$
$$=\frac{1}{\sin\theta\cos\theta}=\csc\theta\sec\theta$$
