Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variable where both sides of the equation are defined.

The Pythagorean identity states that for any angle x, $$\tan^2 x + 1 = \frac{1}{\text{cos}^2 x}$$ or equivalently, $$\text{sin}^2 x + \text{cos}^2 x = 1$$ .

The reciprocal identity for sine states that $$\text{csc} x = \frac{1}{\text{sin} x}$$ .

The reciprocal identity for cosine states that $$\text{sec} x = \frac{1}{\text{cos} x}$$ .

The reciprocal identity for tangent states that $$\text{cot} x = \frac{1}{\text{tan} x}$$ .

The quotient identity states that $$\tan x = \frac{\text{sin} x}{\text{cos} x}$$ .

The quotient identity states that $$\text{cot} x = \frac{\text{cos} x}{\text{sin} x}$$ .

$$\frac{\text{sin} x}{\frac{1}{\text{sin} x}} = \text{sin}^2 x$$ .

The identity states that $$\text{sin}(-x) = -\text{sin} x$$ .

The identity states that $$\text{cos}(-x) = \text{cos} x$$ .