Editing Trigonometric functions (section)

==Basic identities==
Many [[identity (mathematics)|identities]] interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see [[List of trigonometric identities]]. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval {{math|[0, {{pi}}/2]}}, see [[Proofs of trigonometric identities]]). For non-geometrical proofs using only tools of [[calculus]], one may use directly the differential equations, in a way that is similar to that of the [[#Euler's formula and the exponential function|above proof]] of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.

===Parity===
The cosine and the secant are [[even function]]s; the other trigonometric functions are [[odd function]]s. That is:
:<math>\begin{align}
\sin(-x) &=-\sin x\\
\cos(-x) &=\cos x\\
\tan(-x) &=-\tan x\\
\cot(-x) &=-\cot x\\
\csc(-x) &=-\csc x\\
\sec(-x) &=\sec x.
\end{align}</math>

===Periods===
All trigonometric functions are [[periodic function]]s of period {{math|2{{pi}}}}. This is the smallest period, except for the tangent and the cotangent, which have {{pi}} as smallest period. This means that, for every integer {{mvar|k}}, one has
:<math>\begin{array}{lrl}
\sin(x+&2k\pi) &=\sin x \\
\cos(x+&2k\pi) &=\cos x \\
\tan(x+&k\pi) &=\tan x \\
\cot(x+&k\pi) &=\cot x \\
\csc(x+&2k\pi) &=\csc x \\
\sec(x+&2k\pi) &=\sec x.
\end{array}</math>
See [[#Periodicity_and_asymptotes|Periodicity and asymptotes]].

===Pythagorean identity===

The Pythagorean identity, is the expression of the [[Pythagorean theorem]] in terms of trigonometric functions. It is 
:<math>\sin^2 x  + \cos^2 x  = 1</math>.
Dividing through by either <math>\cos^2 x</math> or <math>\sin^2 x</math> gives
:<math>\tan^2 x  + 1  = \sec^2 x</math>
:<math>1  + \cot^2 x  = \csc^2 x</math>
and
:<math>\sec^2 x  + \csc^2 x  = \sec^2 x \csc^2 x</math>.

===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.

===Derivatives and antiderivatives===
The [[derivative]]s of trigonometric functions result from those of sine and cosine by applying the [[quotient rule]]. The values given for the [[antiderivative]]s in the following table can be verified by differentiating them. The number&nbsp;{{mvar|C}} is a [[constant of integration]].

{| class="wikitable" style="text-align: center;"
!<math>f(x)</math> !! <math>f'(x)</math> !! <math display="inline">\int f(x) \, dx</math>
|-
|<math>\sin x</math>||<math>\cos x</math>||<math>-\cos x + C</math>
|-
|<math>\cos x</math>||<math>-\sin x</math>||<math>\sin x + C</math>
|-
|<math>\tan x</math>||<math>\sec^2 x</math>||<math>\ln \left| \sec x \right| + C</math>
|-
|<math>\csc x</math>||<math>-\csc x \cot x</math>||<math>\ln \left| \csc x - \cot x \right| + C</math>
|-
|<math>\sec x</math>||<math>\sec x \tan x</math>||<math>\ln \left| \sec x + \tan x \right| + C</math>
|-
|<math>\cot x</math>||<math>-\csc^2 x</math>||<math>-\ln \left| \csc x \right| + C</math>
|}
Note: For <math>0<x<\pi</math> the integral of <math>\csc x</math> can also be written as <math>-\operatorname{arsinh}(\cot x),</math> and for the integral of <math>\sec x</math> for <math>-\pi/2<x<\pi/2</math> as <math>\operatorname{arsinh}(\tan x),</math> where <math>\operatorname{arsinh}</math> is the [[inverse hyperbolic sine]].

Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:

:<math>
\begin{align}
\frac{d\cos x}{dx} &= \frac{d}{dx}\sin(\pi/2-x)=-\cos(\pi/2-x)=-\sin x \, , \\
\frac{d\csc x}{dx} &= \frac{d}{dx}\sec(\pi/2 - x) = -\sec(\pi/2 - x)\tan(\pi/2 - x) = -\csc x \cot x \, , \\
\frac{d\cot x}{dx} &= \frac{d}{dx}\tan(\pi/2 - x) = -\sec^2(\pi/2 - x) = -\csc^2 x \, .
\end{align}
</math>