====== Relazioni trigonometriche ======

===== Relazione tangente/coseno =====

$$\sin^2 \theta + \cos^2 \theta = 1 \Longrightarrow \tan^2 \theta + 1 = \frac{1}{\cos^2 \theta} \Longrightarrow \, $$

$$\, \Longrightarrow  \cos^2 \theta = \frac{1}{1 + \tan^2 \theta} \Longrightarrow  \cos \theta = \pm \frac{1}{\sqrt{1 + \tan^2 \theta} } $$

Ricapitolando

$$ \cos \theta = \pm \frac{1}{\sqrt{1 + \tan^2 \theta} } $$

===== Relazione tangente/seno =====

$$\sin^2 \theta + \cos^2 \theta = 1 \Longrightarrow 1 + \frac{1}{\tan^2 \theta } = \frac{1}{\sin^2 \theta} \Longrightarrow \, $$

$$\, \Longrightarrow  \frac{1 + \tan^2 \theta}{\tan^2 \theta } = \frac{1}{\sin^2 \theta} \Longrightarrow sin^2 \theta = \frac{\tan^2 \theta }{1 + \tan^2 \theta} \Longrightarrow sin \theta = \pm \frac{\tan \theta }{\sqrt{1 + \tan^2 \theta}} $$

Quindi la relazione cercata è

$$sin \theta = \pm \frac{\tan \theta }{\sqrt{1 + \tan^2 \theta}} $$