There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine, or on the differential equation to which they are solutions.
Elementary trigonometric identities
Definitions
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
Ratio identities
In the case of angles smaller than a right angle, the following identities are direct consequences of above definitions through the division identity
They remain valid for angles greater than 90° and for negative angles.
Or
Complementary angle identities
Two angles whose sum is π/2 radians (90 degrees) are complementary. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining:
Pythagorean identities
Identity 1:
The following two results follow from this and the ratio identities. To obtain the first, divide both sides of by ; for the second, divide by .
Similarly
Identity 2:
The following accounts for all three reciprocal functions.
Proof 2:
Refer to the triangle diagram above. Note that by Pythagorean theorem.
Substituting with appropriate functions -
Rearranging gives:
Angle sum identities
Sine
Draw a horizontal line (the x-axis); mark an origin O. Draw a line from O at an angle above the horizontal line and a second line at an angle above that; the angle between the second line and the x-axis is .
Place P on the line defined by at a unit distance from the origin.
Let PQ be a line perpendicular to line OQ defined by angle , drawn from point Q on this line to point P. OQP is a right angle.
Let QA be a perpendicular from point A on the x-axis to Q and PB be a perpendicular from point B on the x-axis to P. OAQ and OBP are right angles.
Draw R on PB so that QR is parallel to the x-axis.
Now angle (because , making , and finally )
, so
, so
By substituting for and using Symmetry, we also get:
Cosine
Using the figure above,
, so
, so
By substituting for and using Symmetry, we also get:
Also, using the complementary angle formulae,
Tangent and cotangent
From the sine and cosine formulae, we get
Dividing both numerator and denominator by , we get
Subtracting from , using ,
Similarly from the sine and cosine formulae, we get
Then by dividing both numerator and denominator by , we get
Or, using ,
Using ,
Double-angle identities
From the angle sum identities, we get
and
The Pythagorean identities give the two alternative forms for the latter of these:
The angle sum identities also give
It can also be proved using Euler's formula
Squaring both sides yields
But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields
It follows that
.
Expanding the square and simplifying on the left hand side of the equation gives
.
Because the imaginary and real parts have to be the same, we are left with the original identities
,
and also
.
Half-angle identities
The two identities giving the alternative forms for cos 2θ lead to the following equations:
The sign of the square root needs to be chosen properly—note that if 2 is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ.
For the tan function, the equation is:
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:
Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
This also gives:
Similar manipulations for the cot function give: