### Instruction: Simplify $\cot 10 + \tan 5.$

Enter your answer as a trigonometric function evaluated at an integer, such as "sin 7".### Response: I know that $\cot 10 = \frac{\cos 10}{\sin 10}$ and $\tan 5 = \frac{\sin 5}{\cos 5}$, so I can write the expression as $\frac{\cos 10}{\sin 10} + \frac{\sin 5}{\cos 5}.$
To add fractions, I need a common denominator, so I multiply the first fraction by $\frac{\cos 5}{\cos 5}$ and the second fraction by $\frac{\sin 10}{\sin 10}.$
This gives me $\frac{\cos 10 \cos 5}{\sin 10 \cos 5} + \frac{\sin 5 \sin 10}{\cos 5 \sin 10}.$
Now I can combine the numerators and keep the common denominator: $\frac{\cos 10 \cos 5 + \sin 5 \sin 10}{\sin 10 \cos 5}.$
I recognize that the numerator is the same as $\cos (10 - 5)$ by the cosine subtraction formula, so I can simplify it to $\frac{\cos 5}{\sin 10 \cos 5}.$
Finally, I can cancel out the common factor of $\cos 5$ and get $\frac{1}{\sin 10}.$
This is the same as $\csc 10$, the cosecant of 10, which is the reciprocal of the sine of 10.