Half-Angle Identity for Tangent: Proof and Formula

tan(x/2) = +/- √((1 – cosx) / (1 + cosx))

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The identity you provided is known as the half-angle identity for tangent. It states that for any angle x,

tan(x/2) = +/- √((1 – cosx) / (1 + cosx))

Here’s the proof of the identity:

We start with the half-angle formula for sine and cosine:

sin(x/2) = +/- √((1 – cosx) / 2) cos(x/2) = +/- √((1 + cosx) / 2)

Dividing the sine of x/2 by the cosine of x/2 gives us the tangent of x/2, so we have:

tan(x/2) = sin(x/2) / cos(x/2)

Substituting the expressions for sin(x/2) and cos(x/2) into this expression, we get:

tan(x/2) = +/- √((1 – cosx) / 2) / +/- √((1 + cosx) / 2)

Simplifying the fractions, we get:

tan(x/2) = +/- √((1 – cosx) / (1 + cosx))

This is the half-angle identity for tangent.

Note: The +/- sign in the identity depends on the quadrant in which x/2 lies. If x/2 is in the second or third quadrant, the sign is negative; if x/2 is in the first or fourth quadrant, the sign is positive.

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