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$\tan \theta = \frac{\sin \theta}{\cos \theta}$

Prerequisites

• Proof by definition of the functions: sine, cosine, and tangent as the ratios of their respective sides in a right triangle.
• $\sin \theta = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$
• $\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$
• $\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}}$

Proof

Definition of Tangent:

$\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}}$

$\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} \cdot 1$

$\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} \cdot \frac{\mathrm{hypotenuse}}{\mathrm{hypotenuse}}$

$\tan \theta = \frac{ \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}}{\frac{ \mathrm{adjacent}}{\mathrm{hypotenuse}}}$

Definition of Sine and Cosine:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$