Editing Trigonometric functions (section)

===Sum and difference formulas===

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to [[Ptolemy]] (see [[List_of_trigonometric_identities#Angle_sum_and_difference_identities|Angle sum and difference identities]]). One can also produce them algebraically using [[Euler's formula]].
; Sum
:<math>\begin{align}
\sin\left(x+y\right)&=\sin x \cos y + \cos x \sin y,\\[5mu]
\cos\left(x+y\right)&=\cos x \cos y - \sin x \sin y,\\[5mu]
\tan(x + y) &= \frac{\tan x + \tan y}{1 - \tan x\tan y}.
\end{align}</math>
; Difference
:<math>\begin{align}
\sin\left(x-y\right)&=\sin x \cos y - \cos x \sin y,\\[5mu]
\cos\left(x-y\right)&=\cos x \cos y + \sin x \sin y,\\[5mu]
\tan(x - y) &= \frac{\tan x - \tan y}{1 + \tan x\tan y}.
\end{align}</math>

When the two angles are equal, the sum formulas reduce to simpler equations known as the [[double-angle formulae]].

:<math>\begin{align}
\sin 2x &= 2 \sin x \cos x = \frac{2\tan x}{1+\tan^2 x}, \\[5mu]
\cos 2x &= \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x = \frac{1-\tan^2 x}{1+\tan^2 x},\\[5mu]
\tan 2x &= \frac{2\tan x}{1-\tan^2 x}.
\end{align}</math>

These identities can be used to derive the [[product-to-sum identities]].

By setting <math>t=\tan \tfrac12 \theta,</math> all trigonometric functions of <math>\theta</math> can be expressed as [[rational fraction]]s of <math>t</math>:
:<math>\begin{align}
\sin \theta &= \frac{2t}{1+t^2}, \\[5mu]
\cos \theta &= \frac{1-t^2}{1+t^2},\\[5mu]
\tan \theta &= \frac{2t}{1-t^2}.
\end{align}</math>
Together with 
:<math>d\theta = \frac{2}{1+t^2} \, dt,</math>
this is the [[tangent half-angle substitution]], which reduces the computation of [[integral]]s and [[antiderivative]]s of trigonometric functions to that of rational fractions.