Deriving the Double-Angle Identities

Why Double-Angle Identities Follow from Angle Addition

An angle addition formula describes the sine or cosine of a sum of two angles. A double-angle expression is the special case in which those two angles are equal: \(x+x=2x\). This simple substitution produces useful identities for \(\sin(2x)\), \(\cos(2x)\), and, when the relevant denominators are nonzero, \(\tan(2x)\).

The sine and cosine identities are valid for every real \(x\). The tangent identity needs more care: tangent is a quotient, so it is defined only when its cosine denominator is nonzero. Keeping track of those restrictions is part of deriving the identity correctly.

Deriving the Sine Double-Angle Identity

Set \(a=x\) and \(b=x\) in the sine addition formula. Each sine and cosine on the right is then evaluated at \(x\):

The last equality uses the commutativity of multiplication: \(\cos x\sin x=\sin x\cos x\). There are no divisions in this derivation, so the result holds for every real \(x\).

Deriving the Cosine Double-Angle Identities

Now set \(a=b=x\) in the cosine addition formula. This gives the first form directly:

Two other forms follow from the Pythagorean identity \(\sin^2 x+\cos^2 x=1\). To obtain a form involving only \(\cos x\), replace \(\sin^2 x\) with \(1-\cos^2 x\):

Alternatively, replace \(\cos^2 x\) with \(1-\sin^2 x\):

These are not three separate rules about different angles. They are equal expressions for the same quantity. The first comes directly from angle addition; the other two are algebraic consequences of the Pythagorean identity.

Deriving the Tangent Double-Angle Identity

Tangent is defined by \(\tan u=\frac{\sin u}{\cos u}\) when \(\cos u\ne 0\). Therefore \(\tan(2x)\) is defined only if \(\cos(2x)\ne 0\). To express it in terms of \(\tan x\), we also need \(\tan x\) to be defined, which requires \(\cos x\ne 0\).

Under those conditions, use the sine and cosine double-angle identities and divide:

The division by \(\cos^2 x\) is valid because \(\cos x\ne 0\). The final denominator must also be nonzero: \(1-\tan^2 x\ne 0\). With \(\cos x\ne 0\), the cosine identity gives \(\cos(2x)=\cos^2 x(1-\tan^2 x)\), so this last condition is exactly what ensures \(\cos(2x)\ne 0\).

A Complete Derivation in Steps

The derivations can be organized as a short sequence. The domain check belongs specifically to the tangent step because that step involves division.

Worked Examples

A Common Mistake: Ignoring the Denominator

A frequent error is to use the tangent identity whenever \(\tan x\) exists, without checking the denominator \(1-\tan^2 x\). For example, at \(x=\pi/4\), \(\tan x=1\) exists, but the proposed calculation

is not a real number. It does not mean that \(\tan(\pi/2)\) has some value obtained by this calculation. Rather, both the right-hand expression and \(\tan(\pi/2)\) are undefined. The correction is to check the denominator before applying the formula and to use the sine and cosine identities when those quantities are what is needed.

Why These Identities Matter

The identities connect trigonometric values at \(x\) to values at \(2x\). They let a calculation involving a doubled angle be rewritten using the sine, cosine, or tangent of the original angle. The derivation also illustrates a general mathematical habit: algebraic rearrangement is safe only after checking that every division is permitted.

For sine and cosine, setting equal angles in the addition formulas introduces no restrictions. For tangent, the quotient definition creates domain conditions that cannot be dropped. Remembering this distinction makes the identities both useful and precise.