# Trigonometric Identities – Explanation and Examples

Trigonometric identities are equivalence relationships between two expressions involving one or more trigonometric functions that are true for all angles.

These identities help to simplify complicated trigonometric equations. Since it can make the expression easier to differentiate or integrate, this can be useful when dealing with trigonometric functions in calculus.

Before reading on, make sure to review trigonometric functions.

## List of Trigonometric Identities

Technically, trigonometric identities cover definitional identities such as $sine=\frac{opposite}{hypotenuse}$ and conversions between radians, degrees, and gradians. Because of this, there are infinitely many different trigonometric identities covering many different identity types.

When most people talk about trigonometric identities, however, they mean one of the following broader categories of identities.

• Pythagorean Identities – These include $sin^2x+cos^2x=1$ and related identities, such as $sin^2x=1-cos^2x$.
• Reciprocal Identities – One divided by sine is cosecant is one example of a reciprocal identity.
• Reflections, Shifts, and Periodicity – These identities describe relationships related to reflections and shifts of functions. They also include shifts. These identities include $sin(-x)=-sinx$, $sin(x+\pi)=-sinx$, and $sin(x+2\pi)=sinx$.
• Angle Sum and Difference Identities – These identities include $sin(a+b)=sin(a)cos(b)+cos(a)sin(b)$.
• Multiple-Angle Identities – Half-angle, double-angle, and triple-angle identities are the most famous multiple-angle identities.
• Power-Reduction Identities – There are many identities that convert one trigonometric function raised to a power (such as $sin^2x$) into an expression that only involves first-degree functions.
• Product-to-Sum Identities – These identities turn products of trigonometric functions into sums of trigonometric functions. One example is $2cosxcosy = cos(x-y)+cos(x+y)$.

Then there are other, more complicated trigonometric identities such as Lagrange’s Identity that deal with series.

### Basic Trigonometric Identities

Although there are many trigonometric identities, the most common and useful ones are these. Note that this list is not exhaustive as other identities can be derived from those shown. For example, $sin^2x+cos^2x=1$ means that $sin^2x=1-cos^2x$.

#### Pythagorean Identities

Here are the Pythagorean Identities:
• $sin^2x+cos^2x=1$
• $1+tan^2x=sec^2x$
• $1+cot^2x=csc^2x$

#### Reciprocal Identities

Here are the Reciprocal Identities:
• $cscx = \frac{1}{sinx}$
• $secx = \frac{1}{cosx}$
• $cotx = \frac{1}{tanx}$

#### Half-Angle Identities

Here are the Half-angle Identities:
• $sin(\frac{x}{2}) = \pm \sqrt{\frac{1-cosx}{2}}$
• $cos(\frac{x}{2}) = \pm \sqrt{\frac{1+cosx}{2}}$
• $tan(2x) = \frac{2tanx}{1-tan^2x} #### Product-to-Sum Identities Here are the Product-to-Sum Identities: •$sinxsiny = \frac{1}{2}[cos(x-y)-cos(x+y)]
• $cosxcosy = \frac{1}{2}[cos(x-y)+cos(x+y)] •$sinxcosy = \frac{1}{2}[sin(x+y)-sin(x-y)]

#### Sum-to-Product Identities

Here are the Sum-to-Product Identities:
• $sinx+siny = 2sin(\frac{x+y}{2})cos(\frac{x-y}{2})$
• $sinx-siny = 2sin(\frac{x-y}{2})cos(\frac{x+y}{2})$