Proofs of trigonometric identities

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Proofs of trigonometric identities are used to show relations between trigonometric functions. This article will list trigonometric identities and prove them.

Elementary trigonometric identities[edit]


Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.

Referring to the diagram at the right, the six trigonometric functions of θ are:

Ratio identities[edit]

The following identities are trivial algebraic consequences of these definitions and the division identity.
They rely on multiplying or dividing the numerator and denominator of fractions by a variable. Ie,


Complementary angle identities[edit]

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[edit]

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 .


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[edit]


Illustration of the sum formula.

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 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:

Another simple "proof" can be given using Euler's formula known from complex analysis: Euler's formula is:

Although it is more precise to say that Euler's formula entails the trigonometric identities, it follows that for angles and we have:

Also using the following properties of exponential functions:

Evaluating the product:

Equating real and imaginary parts:


Using the figure above,

, so
, so

By substituting for and using Symmetry, we also get:

Also, using the complementary angle formulae,

Tangent and cotangent[edit]

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[edit]

From the angle sum identities, we get


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[edit]

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 π 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[edit]

If half circle (for example, , and are the angles of a triangle),


Miscellaneous -- the triple cotangent identity[edit]

If quarter circle,



Replace each of , , and with their complementary angles, so cotangents turn into tangents and vice versa.


so the result follows from the triple tangent identity.

Prosthaphaeresis identities[edit]

Proof of sine identities[edit]

First, start with the sum-angle identities:

By adding these together,

Similarly, by subtracting the two sum-angle identities,

Let and ,


Substitute and


Proof of cosine identities[edit]

Similarly for cosine, start with the sum-angle identities:

Again, by adding and subtracting

Substitute and as before,


Illustration of the sine and tangent inequalities.

The figure at the right shows a sector of a circle with radius 1. The sector is θ/(2π) of the whole circle, so its area is θ/2.

The area of triangle OAD is AB/2, or sinθ/2. The area of triangle OCD is CD/2, or tanθ/2.

Since triangle OAD lies completely inside the sector, which in turn lies completely inside triangle OCD, we have

This geometric argument applies if 0<θ<π/2. It 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.[2] For the sine function, we can handle other values. If θ>π/2, then θ>1. But sinθ≤1 (because of the Pythagorean identity), so sinθ<θ. So we have

For negative values of θ we have, by symmetry of the sine function


Identities involving calculus[edit]


Sine and angle ratio identity[edit]

Proof: From the previous inequalities, we have, for small angles




Consider the right-hand inequality. Since

Multiply through by

Combining with the left-hand inequality:

Taking to the limit as


Cosine and angle ratio identity[edit]


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[edit]


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[edit]

All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function


We start from

Then we divide this equation by

Then use the substitution , also use the Pythagorean trigonometric identity:

Then we use the identity

See also[edit]


  1. ^ dead link
  2. ^ Richman, Fred (March 1993). "A Circular Argument". The College Mathematics Journal. 24 (2): 160–162. doi:10.2307/2686787. Retrieved 3 November 2012. 


  • E. T. Whittaker and G. N. Watson. A course of modern analysis, Cambridge University Press, 1952