Proofs of trigonometric identities

Proofs of trigonometric identities
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:
Miscellaneous – the triple tangent identity
If half circle (for example and are the angles of a triangle),
Proof:
Miscellaneous – the triple cotangent identity
If quarter circle,
.
Proof:
Replace each of and with their complementary angles, so cotangents turn into tangents and vice versa.
Given
so the result follows from the triple tangent identity.
Sum to product identities
Proof of sine identities
First, start with the sum-angle identities:
By adding these together,
Similarly, by subtracting the two sum-angle identities,
Let and ,
and
Substitute and
Therefore,
Proof of cosine identities
Similarly for cosine, start with the sum-angle identities:
Again, by adding and subtracting
Substitute and as before,
Inequalities
The figure at the right shows a sector of a circle with radius 1. The sector is of the whole circle, so its area is . We assume here that .
The area of triangle is , or . The area of triangle is , or .
Since triangle lies completely inside the sector, which in turn lies completely inside triangle , we have
This geometric argument relies on definitions of arc length and
area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than
a provable property. For the sine function, we can handle other values. If , then . But (because of the Pythagorean identity), so . So we have
For negative values of we have, by the symmetry of the sine function
Hence
and
Identities involving calculus
Preliminaries
Sine and angle ratio identity
In other words, the function sine is differentiable at 0, and its derivative is 1.
Proof: From the previous inequalities, we have, for small angles
,
Therefore
Consider the right-hand inequality. Since
Multiply through by
Combining with the left-hand inequality:
Taking to the limit as
Therefore,
Cosine and angle ratio identity
Proof:
The limits of those three quantities are 1, 0, and 1/2, so the resultant limit is zero.
Cosine and square of angle ratio identity
Proof:
As in the preceding proof,
The limits of those three quantities are 1, 1, and 1/2, so the resultant limit is 1/2.
Proof of compositions of trig and inverse trig functions
All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function
Proof:
We start from
(I)
Then we divide this equation (I) by
(II)
Then use the substitution :
Then we use the identity
(III)
And initial Pythagorean trigonometric identity proofed...
Similarly if we divide this equation (I) by
(II)
Then use the substitution :
Then we use the identity
(III)
And initial Pythagorean trigonometric identity proofed...
(IV)
Let we guess that we have to prove:
(V)
Replacing (V) into (IV) :
So it's true: and guessing statement was true:
Now y can be written as x ; and we have [arcsin] expressed through [arctan]...
Similarly if we seek :...
From :...
And finally we have [arccos] expressed through [arctan]...
See also
List of trigonometric identities
Bhaskara I's sine approximation formula
Generating trigonometric tables
Aryabhata's sine table
Madhava's sine table
Table of Newtonian series
Madhava series
Unit vector (explains direction cosines)
Euler's formula
Notes
References
E. T. Whittaker and G. N. Watson. A Course of Modern Analysis, Cambridge University Press, 1952
Trigonometry
Article proofs