# Rearranging a trigonometric expression

I am working on a problem and obtained the following equation:

$$\tan \beta = \frac{\lambda - \cos \theta}{\sin \theta} \Leftrightarrow \tan \beta = \lambda\cdot \csc \theta-\cot \theta$$

Where $$\lambda$$ is a constant.

Is it possible to simplify the right hand side term into a single trig function? My goal is to have an equation of the form $$\theta = f(\beta)$$

I have tried different trig identities but I only end up making it worse.

Any help is appreciated.

Thank you!

Note that $$\sin \theta\sin\beta+\cos\theta\cos\beta=\lambda\cos\beta$$ and the LHS equivalent to $$\cos(\theta-\beta).$$
The answer to your question is that the inverse equation, $$\theta=f(\beta)$$ has three values:
$$\theta=arccos(\lambda * cos(\beta))$$
$$\theta=arcsin(\lambda *cos(\beta))+\beta+\frac{3 \pi}{2}$$
$$\theta=\beta-arccos(\lambda* cos(\beta))$$