# What Are Double Angle Identities? ••• demaerre/iStock/GettyImages

Once you start doing trigonometry and calculus, you may run into trigonometric functions like sine, cosine, and tangent. Playing trial and error with charts or a calculator to find the answer to trigonometric equations would range from a drawn-out nightmare to totally impossible. The many trig identities and relationships become crucial when solving for these trigonometric ratios. The double-angle identities are special instances of what's known as a compound formula, which breaks functions of the forms (‌A + B‌) or (‌A - B‌) down into functions of either ‌A‌ or ‌B‌.

There are many other trigonometric identities that you might recognize. The pythagorean identity for a right triangle, half-angle formulas, sum formula, and difference identities are all very useful relationships.

#### TL;DR (Too Long; Didn't Read)

These identities can also be called theorems, which refers to a slightly different item in math, but functionally they describe the same thing. Some of these identities also have equivalent names (half-angle identities, sum identities, addition formulas, etc.).

There are three double-angle identities, one each for the sine, cosine and tangent functions. The sine and cosine functions can both be written with multiple special cases.

## The Double-Angle Identities for Sine

Here are the two ways of writing the double-angle identity for the sine function:

\begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta) \\ &= \frac{2\tan{\theta}}{1 + \tan^2\theta} \end{align*}

## The Double-Angle Identities for Cosine

There are even more ways of writing the double-angle identity for cosine:

\begin{align*} \cos{2\theta} &= \cos^2(\theta) - \sin^2(\theta) \\ &= 2\cos^2(\theta) - 1 \\ &= 1 - 2\sin^2(\theta) \\ &= \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)} \end{align*}

## The Double-Angle Identity for Tangent

There is just one practical way to write the double-angle identity for the tangent function:

\begin{align*} \tan(2\theta) &= \frac{2\tan(\theta)}{1-\tan^2(\theta)} \end{align*}

## Using Double-Angle Identities

There are numerous trigonometric expressions and scenarios to solve. These formulas will often use theta (‌Θ), x‌ or alpha (‌ɑ‌), but regardless of the variable, the double angle formula will be able to find an equivalent exact value for a suitable situation.

#### Tips

• The variable ‌x‌ was commonly used in all algebraic problems in early algebra and precalculus, but when we look at sin ‌x‌ cos ‌x‌ tan x this is equivalent to using theta or alpha, such as sin α cos α tan α. All variables simply keep track of possible values and their position on each side of the equation.

## Example 1

Suppose we want to find an equivalent statement in terms of sin ‌θ‌ and cos ‌θ‌ to the following trig function:

\cos2x + \sin2x

We can use the double angle formulas to reduce this expression to only use sin ‌θ‌ and cos ‌θ:

\begin{align*} (2\cos^2(x) - 1) + (2\sin(x)\cos(x)) &= (2\cos^2(x) + 2\sin(x)\cos(x)) - 1 \\ & = 2\cos(x)(\cos(x) + \sin(x)) - 1 \end{align*}

## Example 2

Simplify the following expressions:

2\cos^2(32) -1 \\ \text{} \\ 2\sin α \cos α \ \text{ where } α = \frac{1}{2}\beta

Using the cosine double-angle identity on the first expression we can just use the value of cos to represent the same value:

\begin{align*} 2\cos^2(32)-1 &=\cos{2 \times 32} \\ &=\cos{64} \end{align*}

We can apply a similar process to the second expression using the sine double-angle identity:

\begin{align*} 2\sin{α}\cos{α} &= \sin{2 \times α} \\ &= \sin{2 \times \frac{\beta}{2}} \\ &= \sin{\beta} \end{align*}

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