[Math] If $ \sin\theta + \cos\theta = \frac 1 2$, what does $\tan\theta + \cot\theta$ equal

Question

A SAT II question asks:

If $ \sin\theta + \cos\theta = \dfrac 1 2$, what does $\tan\theta + \cot\theta$ equal?

Which identity would I need to solve this?

Answer 1

Hint

$$\sin\theta+\cos\theta=\frac{1}{2} \implies \left( \sin\theta+\cos\theta \right)^2 = \frac{1}{4} \iff \color{blue}{\cos\theta\sin\theta} = \cdots$$ and $$\tan\theta+\cot\theta = \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta} = \frac{\cos^2\theta+\sin^2\theta}{\cos\theta\sin\theta}= \frac{1}{\color{blue}{\cos\theta\sin\theta}} = \cdots$$