What Are Double Angle Identities?

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.

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.

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

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

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

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

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

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

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.

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:‌

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

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:

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