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:**

$\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$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**

$\sin^2 \theta + \cos^2 \theta \equiv 1$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 $\sin^2(x) = 1 - \cos^2(x):$

$= \frac{\cos^2(x) - (1 - \cos^2(x))}{\cos(x)} + \frac{1}{\cos(x)}$$= \frac{2\cos^2(x) - 1}{\cos(x)} + \frac{1}{\cos(x)}$$= \frac{2\cos^2(x)}{\cos(x)}$$= 2\cos(x)$$\therefore \frac{\cos^2(x) - \sin^2(x)}{\cos(x)} + \frac{1}{\cos(x)} \equiv 2 \cos(x)$Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.