In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Notation: With trigonometric functions, we define functions sin2, cos2, etc., such that sin2(x) = (sin(x))2.
These are most easily shown from the unit circle:
The quickest way to prove these is Euler's formula.
The tangent formula follows from the other two. A geometric proof of the sin(x+y) identity is given at the end of this article.
These can be shown by substituting in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre's formula with .
If Tn is the nth Chebyshev polynomial then
From the Definitions
Periodicity, Symmetry and Shifts
For some purposes it is important to know that any linear combination of sine waves of the same period but different phase shifts is also a sine wave with the same period, but a different phase shift. In other words, we have
where
From the Pythagorean Theorem
Addition/Subtraction Theorems
whereDouble-Angle Formulas
Multiple-Angle Formulas
De Moivre's formula:
Solve the third and fourth double angle formula for cos2(x) and sin2(x).
Substitute x/2 for x in the power reduction formulas, then solve for cos(x/2) and sin(x/2).
These can be proven by expanding their right-hand-sides using the addition theorems.
Replace x by (x + y) / 2 and y by (x – y) / 2 in the Product-to-Sum formulas.
Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity:
Power-Reduction Formulas
Half-Angle Formulas
Multiply tan(x/2) by 2cos(x/2) / ( 2cos(x/2)) and substitute sin(x/2) / cos(x/2) for tan(x/2). The numerator is then sin(x) via the double angle formula, and the denominator is 2cos2(x/2) - 1 + 1 which is cos(x) + 1 by the double angle formulae. The second formula comes from the first formula multiplied by sin(x) / sin(x) and simplified using the pythagorean trig identity.Products to Sums
Sums to Products
Inverse Trigonometric Functions
Identities with no variables
However, this identity is a special case of one that does contain a variable:
The following are perhaps not as readily generalized to identities with variables in them:
Degree-measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: They are the integers less than 21/2 that have no prime factors in common with 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials; the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above.
In calculus it is essential that angles that are arguments to trigonometric functions be measured in radians; if they are measured in degrees or any other units, then the relations stated below fail. If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying that
Calculus
and
and then using the limit definition of the derivative and the addition theorems; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term.
Abstract point of view
Since the circle is an algebraic curve of genus 0, one expects the 'circular functions' to be reducible to rational functions. This is known classically, by systematically using the tan-half-angle formulae to write the sine and cosine functions in terms of a new variable t.
