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

Enter your answer as a trigonometric function evaluated at an integer, such as "sin 7".### Response: To simplify this expression, I want to use some trigonometric identities that relate cotangent and tangent to other functions.
One identity that comes to mind is $\cot x = \frac{\cos x}{\sin x}$, which can be derived from the definition of cotangent as the reciprocal of tangent.
Using this identity, I can rewrite the first term as $\frac{\cos 10}{\sin 10}$.
Another identity that I can use is $\tan x = \frac{\sin x}{\cos x}$, which is also derived from the definition of tangent as the ratio of sine and cosine.
Using this identity, I can rewrite the second term as $\frac{\sin 5}{\cos 5}$.
Now I have the expression $\frac{\cos 10}{\sin 10} + \frac{\sin 5}{\cos 5}$.
To add these fractions, I need to find a common denominator.
I notice that the product of the denominators is $\sin 10 \cos 5$, so I can use that as the common denominator.
To get the common denominator, I need to multiply the first fraction by $\frac{\cos 5}{\cos 5}$ and the second fraction by $\frac{\sin 10}{\sin 10}$.
This gives me $\frac{\cos 10 \cos 5}{\sin 10 \cos 5} + \frac{\sin 5 \sin 10}{\sin 10 \cos 5}$.
Now I can combine the fractions and write the numerator as a single expression.
I notice that the numerator has the form $\cos a \cos b + \sin a \sin b$, where $a = 10$ and $b = 5$.
This is another trigonometric identity that I can use, namely $\cos a \cos b + \sin a \sin b = \cos (a - b)$.
Using this identity, I can simplify the numerator to $\cos (10 - 5)$, which is $\cos 5$.
Therefore, the final expression is $\frac{\cos 5}{\sin 10 \cos 5}$.
I can further simplify this by canceling out one factor of $\cos 5$ from the numerator and denominator.
This leaves me with $\frac{1}{\sin 10}$.