Sum, Difference and Double-Angle Formulas

Sum and difference

Sine of a sum

\sin (a + b) = \sin a \cos b + \cos a \sin b

Sine of a difference

\sin (a - b) = \sin a \cos b - \cos a \sin b

Cosine of a sum

\cos (a + b) = \cos a \cos b - \sin a \sin b

Cosine of a difference

\cos (a - b) = \cos a \cos b + \sin a \sin b

Tangent of a sum

\tan (a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}

Tangent of a difference

\tan (a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}

Sine

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

Cosine

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

Cosine, in tangent

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

Tangent

\tan 2 x = \frac{2 \tan x}{1 - \tan^{2} x}

Sine times cosine

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

Sine

\sin 3 x = 3 \sin x - 4 \sin^{3} x

Cosine

\cos 3 x = 4 \cos^{3} x - 3 \cos x

Tangent

\tan 3 x = \frac{3 \tan x - \tan^{3} x}{1 - 3 \tan^{2} x}

Sine squared

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

Cosine squared

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

Tangent squared

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

One minus cosine

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

One plus cosine

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

Sine

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

sign from the quadrant of x/2

Cosine

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

sign from the quadrant of x/2

Tangent

\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x}

Tangent, root form

\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}

sign from the quadrant of x/2

Sine times sine

\sin a \sin b = \frac{1}{2} [\cos (a - b) - \cos (a + b)]

Cosine times cosine

\cos a \cos b = \frac{1}{2} [\cos (a - b) + \cos (a + b)]

Sine times cosine

\sin a \cos b = \frac{1}{2} [\sin (a + b) + \sin (a - b)]

Sum of sines

\sin a + \sin b = 2 \sin \frac{a + b}{2} \cos \frac{a - b}{2}

Difference of sines

\sin a - \sin b = 2 \cos \frac{a + b}{2} \sin \frac{a - b}{2}

Sum of cosines

\cos a + \cos b = 2 \cos \frac{a + b}{2} \cos \frac{a - b}{2}

Difference of cosines

\cos a - \cos b = - 2 \sin \frac{a + b}{2} \sin \frac{a - b}{2}

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