Trigonometric identities are crucial in mathematics, offering insights into the relationships between angles and their trigonometric functions. These identities are indispensable for solving complex problems in trigonometry, calculus, and beyond.

Fundamental Trigonometric Identities

The fundamental trigonometric identities include $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and $\sin^2(\theta) + \cos^2(\theta) = 1$. These are derived from the unit circle and the definitions of sine and cosine functions.

1. Tangent Identity:

This identity expresses tangent in terms of sine and cosine.

Examples

Simplify $\tan \theta$ using the tangent identity when $\sin \theta = \frac{3}{5}$ and $\cos \theta = \frac{4}{5}$.

Solution:

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}$

2. Pythagorean Identity

This identity reveals a fundamental relation between sine and cosine.

Examples

Prove the identity $\frac{\cos^2(x) - \sin^2(x)}{\cos(x)} + \frac{1}{\cos(x)} \equiv 2 \cos(x)$.

Solution:

Using $\sin^2(x) = 1 - \cos^2(x):$