Solution

In order to simplify, I will use the method of converting everything to Sine and Cosine. Note that this method is not always the most efficient method for solving and sometimes it is better to leave the other trig functions in place. However, it is a "safe" method in that usually you can make some progress in rewriting the expression, and it comes up as the best method in a large number of circumstances. \[ \solve{ \cos\theta\cot\theta+\sin\theta &=& \cos\theta\left(\frac{\cos\theta}{\sin\theta}\right)+\sin\theta\\ &=&\frac{\cos^2\theta}{\sin\theta}+\sin\theta\\ &=&\frac{\cos^2\theta}{\sin\theta}+\frac{\sin^2\theta}{\sin\theta}\\ &=&\frac{\cos^2\theta+\sin^2\theta}{\sin\theta}\\ &=&\frac{1}{\sin\theta}\\ &=&\csc\theta } \] Note the final step: when simplifying the trig expression, we want to eliminate any fractions we can, even if that means using the reciprocal function!