Proofs of trigonometric identities

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

Elementary trigonometric identities

Definitions

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:

$\sin \theta = \frac {\mathrm{opposite}}{\mathrm{hypotenuse}} = \frac {a}{h}$

$\cos \theta = \frac {\mathrm{adjacent}}{\mathrm{hypotenuse}} = \frac {b}{h}$

$\tan \theta = \frac {\mathrm{opposite}}{\mathrm{adjacent}} = \frac {a}{b}$

$\cot \theta = \frac {\mathrm{adjacent}}{\mathrm{opposite}} = \frac {b}{a}$

$\sec \theta = \frac {\mathrm{hypotenuse}}{\mathrm{adjacent}} = \frac {h}{b}$

$\csc \theta = \frac {\mathrm{hypotenuse}}{\mathrm{opposite}} = \frac {h}{a}$

Ratio identities

The following identities are trivial algebraic consequences of these definitions and the division identity.
c is whatever value (not necessarily trigonometric), only to understand the simple demonstrations above. That is because not appear in the graph.

$\frac {a}{b}= \frac {\left(\frac {a}{c}\right)} {\left(\frac {b}{c}\right) }.$

$\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} = \frac { \left( \frac{\mathrm{opposite}}{\mathrm{hypotenuse}} \right) } { \left( \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\right) } = \frac {\sin \theta} {\cos \theta}.$

$\cot \theta = \frac {\cos \theta}{\sin \theta}.$

$\cot \theta =\frac{\mathrm{adjacent}}{\mathrm{opposite}} = \frac { \left( \frac{\mathrm{adjacent}}{\mathrm{adjacent}} \right) } { \left( \frac {\mathrm{opposite}}{\mathrm{adjacent}} \right) } = \frac {1}{\tan \theta}.$

$\sec \theta = \frac {1}{\cos \theta}$

$\csc \theta = \frac {1}{\sin \theta}$

$\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} = \frac{\left(\frac{\mathrm{opposite} \times \mathrm{hypotenuse}}{\mathrm{opposite} \times \mathrm{adjacent}} \right) } { \left( \frac {\mathrm{adjacent} \times \mathrm{hypotenuse}} {\mathrm{opposite} \times \mathrm{adjacent} } \right) } = \frac{\left( \frac{\mathrm{hypotenuse}}{\mathrm{adjacent}} \right)} { \left( \frac{\mathrm{hypotenuse}}{\mathrm{opposite}} \right)} = \frac {\sec \theta}{\csc \theta}.$

$\cot \theta = \frac {\csc \theta}{\sec \theta}.$

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:

$\sin\left( \pi/2-\theta\right) = \cos \theta$

$\cos\left( \pi/2-\theta\right) = \sin \theta$

$\tan\left( \pi/2-\theta\right) = \cot \theta$

$\cot\left( \pi/2-\theta\right) = \tan \theta$

$\sec\left( \pi/2-\theta\right) = \csc \theta$

$\csc\left( \pi/2-\theta\right) = \sec \theta$

Pythagorean identities

Identity 1:

$\sin^2(x) + \cos^2(x) = 1\,$

Proof 1:

Refer to the triangle diagram above. Note that $a^2+b^2=h^2$ by Pythagorean theorem.

$\sin^2(x) + \cos^2(x) = \frac{a^2}{h^2} + \frac{b^2}{h^2} = \frac{a^2+b^2}{h^2} = \frac{h^2}{h^2} = 1.\,$

The following two results follow from this and the ratio identities. To obtain the first, divide both sides of $\sin^2(x) + \cos^2(x) = 1$ by $\cos^2(x)$; for the second, divide by $\sin^2(x)$.

$\tan^2(x) + 1\ = \sec^2(x)$

$\sec^2(x) - \tan^2(x) = 1\$

Similarly

$1\ + \cot^2(x) = \csc^2(x)$

$\csc^2(x) - \cot^2(x) = 1\$

Proof 2:

Differentiating the left-hand side of the identity yields:

$2 \sin x \cdot \cos x - 2 \sin x \cdot \cos x = 0$

Integrating this shows that the original identity is equal to a constant, and this constant can be found by plugging in any arbitrary value of x.

Identity 2:

The following accounts for all three reciprocal functions.

$\csc^2(x) + \sec^2(x) - \cot^2(x) = 2\ + \tan^2(x)$

Proof 1:

Refer to the triangle diagram above. Note that $a^2+b^2=h^2$ by Pythagorean theorem.

$\csc^2(x) + \sec^2(x) = \frac{h^2}{a^2} + \frac{h^2}{b^2} = \frac{a^2+b^2}{a^2} + \frac{a^2+b^2}{b^2} = 2\ + \frac{b^2}{a^2} + \frac{a^2}{b^2}$

Substituting with appropriate functions -

$2\ + \frac{b^2}{a^2} + \frac{a^2}{b^2} = 2\ + \tan^2(x)+ \cot^2(x)$

Rearranging gives:

$\csc^2(x) + \sec^2(x) - \cot^2(x) = 2\ + \tan^2(x)$

Angle sum identities

Sine

Illustration of the sum formula.

Draw the angles α and β. Place P on the line defined by α + β at unit distance from the origin.

Let PQ be a perpendicular from P to the line defined by the angle α. OQP is a right angle.

Let QA be a perpendicular from Q to the x axis, and PB be a perpendicular from P to the x axis. OAQ is a right angle.

Draw QR parallel to the x-axis. Now angle RPQ = α (because $RPQ = \tfrac{\pi}{2} - RQP = \tfrac{\pi}{2} - (\tfrac{\pi}{2} - RQO) = RQO = \alpha$)

$OP = 1\,$

$PQ = \sin \beta\,$

$OQ = \cos \beta\,$

$\frac{AQ}{OQ} = \sin \alpha\,$, so $AQ = \sin \alpha \cos \beta\,$

$\frac{PR}{PQ} = \cos \alpha\,$, so $PR = \cos \alpha \sin \beta\,$

$\sin (\alpha + \beta) = PB = RB+PR = AQ+PR = \sin \alpha \cos \beta + \cos \alpha \sin \beta\,$

By substituting $-\beta$ for $\beta$ and using Symmetry, we also get:

$\sin (\alpha - \beta) = \sin \alpha \cos -\beta + \cos \alpha \sin -\beta\,$

$\sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\,$

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

$e^{i\varphi}=\cos \varphi +i \sin \varphi$

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

$e^{i (\alpha + \beta)} = \cos (\alpha +\beta) + i \sin(\alpha +\beta)$

Also using the following properties of exponential functions:

$e^{i(\alpha + \beta)} = e^{i \alpha} e^{i\beta}= (\cos \alpha +i \sin \alpha) (\cos \beta + i \sin \beta)$

Evaluating the product:

$e^{i(\alpha + \beta)} = (\cos \alpha \cos \beta - \sin \alpha \sin \beta)+i(\sin \alpha \cos \beta + \sin \beta \cos \alpha)$

This will only be equal to the previous expression we got, if the imaginary and real parts are equal respectively. Hence we get:

$\cos (\alpha +\beta)=\cos \alpha \cos \beta - \sin \alpha \sin \beta$

$\sin (\alpha +\beta)=\sin \alpha \cos \beta + \sin \beta \cos \alpha$

Cosine

Using the figure above,

$OP = 1\,$

$PQ = \sin \beta\,$

$OQ = \cos \beta\,$

$\frac{OA}{OQ} = \cos \alpha\,$, so $OA = \cos \alpha \cos \beta\,$

$\frac{RQ}{PQ} = \sin \alpha\,$, so $RQ = \sin \alpha \sin \beta\,$

$\cos (\alpha + \beta) = OB = OA-BA = OA-RQ = \cos \alpha \cos \beta\ - \sin \alpha \sin \beta\,$

By substituting $-\beta$ for $\beta$ and using Symmetry, we also get:

$\cos (\alpha - \beta) = \cos \alpha \cos - \beta\ - \sin \alpha \sin - \beta\,$

$\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\,$

Also, using the complementary angle formulae,

$\cos (\alpha + \beta) = \sin\left( \pi/2-(\alpha + \beta)\right) = \sin\left( (\pi/2-\alpha) - \beta\right)\,$

$= \sin\left( \pi/2-\alpha\right) \cos \beta - \cos\left( \pi/2-\alpha\right) \sin \beta\,$

$= \cos \alpha \cos \beta - \sin \alpha \sin \beta\,$

Tangent and cotangent

From the sine and cosine formulae, we get

$\tan (\alpha + \beta) = \frac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)}\,$

$= \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta}\,$

Dividing both numerator and denominator by cos α cos β, we get

$\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\,$

$\tan (\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\,$

Similarly from the sine and cosine formulae, we get

$\cot (\alpha + \beta) = \frac{\cos (\alpha + \beta)}{\sin (\alpha + \beta)}\,$

$= \frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\sin \alpha \cos \beta + \cos \alpha \sin \beta}\,$

Then by dividing both numerator and denominator by sin α sin β, we get

$\cot (\alpha + \beta) = \frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}\,$

$\cot (\alpha - \beta) = \frac{\cot \alpha \cot \beta + 1}{\cot \beta - \cot \alpha}\,$

Double-angle identities

From the angle sum identities, we get

$\sin (2 \theta) = 2 \sin \theta \cos \theta\,$

and

$\cos (2 \theta) = \cos^2 \theta - \sin^2 \theta\,$

The Pythagorean identities give the two alternative forms for the latter of these:

$\cos (2 \theta) = 2 \cos^2 \theta - 1\,$

$\cos (2 \theta) = 1 - 2 \sin^2 \theta\,$

The angle sum identities also give

$\tan (2 \theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} = \frac{2}{\cot \theta - \tan \theta}\,$

$\cot (2 \theta) = \frac{\cot^2 \theta - 1}{2 \cot \theta} = \frac{\cot \theta - \tan \theta}{2}\,$

It can also be proved using Euler's formula

$e^{i \varphi}=\cos \varphi +i \sin \varphi$

Squaring both sides yields

$e^{i 2\varphi}=(\cos \varphi +i \sin \varphi)^{2}$

But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields

$e^{i 2\varphi}=\cos 2\varphi +i \sin 2\varphi$

It follows that

$(\cos \varphi +i \sin \varphi)^{2}=\cos 2\varphi +i \sin 2\varphi$.

Expanding the square and simplifying on the left hand side of the equation gives

$i(2 \sin \varphi \cos \varphi) + \cos^2 \varphi - \sin^2 \varphi\ = \cos 2\varphi +i \sin 2\varphi$.

Because the imaginary and real parts have to be the same, we are left with the original identities

$\cos^2 \varphi - \sin^2 \varphi\ = \cos 2\varphi$,

and also

$2 \sin \varphi \cos \varphi = \sin 2\varphi$.

Half-angle identities

The two identities giving the alternative forms for cos 2θ lead to the following equations:

$\cos \frac{\theta}{2} = \pm\, \sqrt\frac{1 + \cos \theta}{2},\,$

$\sin \frac{\theta}{2} = \pm\, \sqrt\frac{1 - \cos \theta}{2}.\,$

The sign of the square root needs to be chosen properly—note that if 2π 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:

$\tan \frac{\theta}{2} = \pm\, \sqrt\frac{1 - \cos \theta}{1 + \cos \theta}.\,$

Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:

$\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}.\,$

Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:

$\tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta}.\,$

This also gives:

$\tan \frac{\theta}{2} = \csc \theta - \cot \theta.\,$

Similar manipulations for the cot function give:

$\cot \frac{\theta}{2} = \pm\, \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} = \frac{1 + \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 - \cos \theta} = \csc \theta + \cot \theta.\,$

Prosthaphaeresis identities

• $\sin \theta \pm \sin y = 2 \sin \frac{\theta\pm y}2 \cos \frac{\theta\mp y}2$

• $\cos \theta + \cos y = 2 \cos \frac{\theta+y}2 \cos \frac{\theta-y}2$

• $\cos \theta - \cos y = -2 \sin \frac{\theta+y}2 \sin \frac{\theta-y}2$

Inequalities

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.

$OA = OD = 1\,$

$AB = \sin \theta\,$

$CD = \tan \theta\,$

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

$\sin \theta < \theta < \tan \theta\,$

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. 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

$\frac{\sin \theta}{\theta} < 1\ \ \ \mathrm{if}\ \ \ 0 < \theta\,$

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

$\frac{\sin \theta}{\theta} = \frac{\sin (-\theta)}{-\theta} < 1\,$

Hence

$\frac{\sin \theta}{\theta} < 1\ \ \ \mathrm{if}\ \ \ \theta \ne 0\,$

$\frac{\tan \theta}{\theta} > 1\ \ \ \mathrm{if}\ \ \ 0 < \theta < \frac{\pi}{2}\,$

Identities involving calculus

Preliminaries

$\lim_{\theta \to 0}{\sin \theta} = 0\,$

$\lim_{\theta \to 0}{\cos \theta} = 1\,$

These can be seen from looking at the diagrams.

Sine and angle ratio identity

$\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1$

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

$\sin \theta < \theta < \tan \theta\,$, so

$\frac{\sin \theta}{\theta} < 1 < \frac{\tan \theta}{\theta}\,$, so

$\frac{\sin \theta}{\theta \cos \theta} > 1\,$, or

$\frac{\sin \theta}{\theta} > \cos \theta\,$, so

$\cos \theta < \frac{\sin \theta}{\theta} < 1\,$, but

$\lim_{\theta \to 0}{\cos \theta} = 1\,$, so

$\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1$

Cosine and angle ratio identity

$\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta} = 0\,$

Proof:

$\frac{1 - \cos \theta}{\theta} = \frac{1 - \cos^2 \theta}{\theta (1 + \cos \theta)}\,$

$= \frac{\sin^2 \theta}{\theta (1 + \cos \theta)}\,$

$= \frac{\sin \theta}{\theta} \times \sin \theta \times \frac{1}{1 + \cos \theta}.\,$

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

$\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2} = \frac{1}{2}$

Proof:

As in the preceding proof,

$\frac{1 - \cos \theta}{\theta^2} = \frac{\sin \theta}{\theta} \times \frac{\sin \theta}{\theta} \times \frac{1}{1 + \cos \theta}.\,$

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

$\sin[\arctan(x)]=\frac{x}{\sqrt{1+x^2}}$

Proof:

We start from

$\sin^2\theta+\cos^2\theta=1$

Then we divide this equation by $\cos^2\theta$

$\cos^2\theta=\frac{1}{\tan^2\theta+1}$

Then use the substitution $\theta=arctan(x)$, also use the Pythagorean trigonometric identity:

$1-\sin^2[\arctan(x)]=\frac{1}{\tan^2[\arctan(x)]+1}$

Then we use the identity $\tan[\arctan(x)]\equiv x$

$\sin[\arctan(x)]=\frac{x}{\sqrt{x^2+1}}$

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

References

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