Half Angle Identities

First, write two of the Double-Angle Identities for cosine.

cos 2 x cos 2 x = 1 - 2 sin^{2} x = 2 cos^{2} x - 1

Sine Identity

Start by solving the first identity written above for

sin x .

cos 2 x = 1 - 2 sin^{2} x

▼

Solve for

sin x

SubEqn

$LHS - 1 = RHS - 1$

cos 2 x - 1 = - 2 sin^{2} x

DivEqn

$LHS / (- 2) = RHS / (- 2)$

\frac{cos 2 x - 1}{- 2} = sin^{2} x

RearrangeEqn

Rearrange equation

sin^{2} x = \frac{cos 2 x - 1}{- 2}

MoveNegDenomToFrac

Put minus sign in front of fraction

sin^{2} x = - \frac{cos 2 x - 1}{2}

DistrNegSignSwap

$- (b - a) = a - b$

sin^{2} x = \frac{1 - cos 2 x}{2}

SqrtEqn

$LHS = RHS$

sin x = \pm \frac{1 - cos 2 x}{2}

Next, substitute

\frac{θ}{2}

for

x

to obtain the half-angle identity for the sine.

sin \frac{θ}{2} sin \frac{θ}{2} = \pm \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{2} ⇓ = \pm \frac{1 - cos θ}{2} ✓

Cosine Identity

Start by solving the second identity written at the beginning for

cos x .

cos 2 x = 2 cos^{2} x - 1

▼

Solve for

cos x

AddEqn

$LHS + 1 = RHS + 1$

1 + cos 2 x = 2 cos^{2} x

DivEqn

$LHS / 2 = RHS / 2$

\frac{1 + cos 2 x}{2} = cos^{2} x

RearrangeEqn

Rearrange equation

cos^{2} x = \frac{1 + cos 2 x}{2}

SqrtEqn

$LHS = RHS$

cos x = \pm \frac{1 + cos 2 x}{2}

Next, substitute

\frac{θ}{2}

for

x

to obtain the half-angle identity for the cosine.

cos \frac{θ}{2} cos \frac{θ}{2} = \pm \frac{1 + cos ( 2 \cdot \frac{θ}{2} )}{2} ⇓ = \pm \frac{1 + cos θ}{2} ✓

Tangent Identity

To derive the tangent identity, start by recalling the definition of the tangent ratio.

tan x = \frac{sin x}{cos x}

Next, substitute

\frac{θ}{2}

for

x .

tan \frac{θ}{2} = \frac{sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )}

Finally, substitute the half-identities for the sine and cosine into the equation above and simplify the right-hand side.

tan \frac{θ}{2} = \frac{sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )}

SubstituteII

$sin \frac{θ}{2} = \pm \frac{1 - c o s θ}{2}$ , $cos \frac{θ}{2} = \pm \frac{1 + c o s θ}{2}$

tan \frac{θ}{2} = \frac{\pm \frac{1 - c o s θ}{2}}{\pm \frac{1 + c o s θ}{2}}

▼

Simplify right-hand side

DivSqrt

$\frac{a}{b} = \frac{a}{b}$

tan \frac{θ}{2} = \pm \frac{\frac{1 - c o s θ}{2}}{\frac{1 + c o s θ}{2}}

DivFracByFracD

$\frac{a / b}{c / d} = \frac{a}{b} \cdot \frac{d}{c}$

tan \frac{θ}{2} = \pm \frac{1 - cos θ}{2} \cdot \frac{2}{1 + cos θ}

CrossCommonFac

Cross out common factors

tan \frac{θ}{2} = \pm \frac{1 - cos θ}{2} \cdot \frac{2}{1 + cos θ}

CancelCommonFac

Cancel out common factors

tan \frac{θ}{2} = \pm \frac{1 - cos θ}{1} \cdot \frac{1}{1 + cos θ}

MultFrac

Multiply fractions

tan \frac{θ}{2} = \pm \frac{1 - cos θ}{1 + cos θ} ✓

Half-Angle Identities

Proof

Sine Identity

Cosine Identity

Tangent Identity

Extra

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