Trigonometry is a branch of Mathematics dealing with the angle and the sides of right-angled triangles. There are several mathematical formulas in trigonometry which are generally divided into two parts: trigonometric identity or trigonometric ratios. Trigonometric identities are the equations based on the laws of triangle, they relate the side and the angle of the triangle. These are the fundamental identities which will be used for solving trigonometric problems in various applications.

There are basically six trigonometric functions; sine, cosine, tangent, cosecant, secant, and cotangent. These identities relate all these functions with each other and deduce several other relationships with different operations of the angle of the triangle. There are many identities but, in this section, these are divided into two parts: one consists of important identities which one must remember. They are more commonly used for solving trigonometric problems and the other one is second string trigonometric identities. These are not common as they have lesser number of applications and can be used only when needed.

Important trigonometric identities

These are very basic and important trigonometric identities. These identities have several applications.

Defining relationship between various trigonometric functions.

\begin{array}{ll}\tan \theta=\frac{\sin \theta}{\cos \theta} & \sec \theta=\frac{1}{\cos \theta} \\ \cot \theta=\frac{1}{\tan \theta}=\frac{\cos \theta}{\sin \theta} & \csc \theta=\frac{1}{\sin \theta}\end{array}

tanÎ¸=

cosÎ¸

sinÎ¸

â€‹

cotÎ¸=

tanÎ¸

1

â€‹

=

sinÎ¸

cosÎ¸

â€‹

â€‹

secÎ¸=

cosÎ¸

1

â€‹

cscÎ¸=

sinÎ¸

1

â€‹

â€‹

Pythagorean identities: These are one of the most important identities derived by the Pythagoras theorem of the right-angled triangle.

\sin ^{2} \theta+\cos ^{2} \theta=1sin

2

Î¸+cos

2

Î¸=1

\tan ^{2} \theta+1=\sec ^{2} \thetatan

2

Î¸+1=sec

2

Î¸

1+\cot ^{2} \theta=\csc ^{2} \theta1+cot

2

Î¸=csc

2

Î¸

Complements identities: These identities represent the trigonometric functions into their co-functions by using complementary angle.

\sin \theta=\cos \left(\frac{\pi}{2}-\theta\right)sinÎ¸=cos(

2

Ï€

â€‹

âˆ’Î¸)

\tan \theta=\cot \left(\frac{\pi}{2}-\theta\right)tanÎ¸=cot(

2

Ï€

â€‹

âˆ’Î¸)

\sec \theta=\csc \left(\frac{\pi}{2}-\theta\right)secÎ¸=csc(

2

Ï€

â€‹

âˆ’Î¸)

Periodicity identity: In these identities; cosine, sine, cosecant, and secant are having 2pi while tangent and cotangent are having a period of pi.

\sin (\theta+2 \pi)=\sin \thetasin(Î¸+2Ï€)=sinÎ¸

\cos (\theta+2 \pi)=\cos \thetacos(Î¸+2Ï€)=cosÎ¸

\tan (\theta+\pi)=\tan \thetatan(Î¸+Ï€)=tanÎ¸

Negative angle identities: We know that sine, cosecant, cotangent, and tangent are all odd functions whereas secant and cosine are both even functions.

\sin (-\theta)=-\sin \thetasin(âˆ’Î¸)=âˆ’sinÎ¸

\cos (-\theta)=\cos \thetacos(âˆ’Î¸)=cosÎ¸

\tan (-\theta)=-\tan \thetatan(âˆ’Î¸)=âˆ’tanÎ¸

Sum and difference identities: These identities are also called Ptolemyâ€™s identities which provide the sum and difference formulas for trigonometric functions.

\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \betasin(Î±+Î²)=sinÎ±cosÎ²+cosÎ±sinÎ²

\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \betacos(Î±+Î²)=cosÎ±cosÎ²âˆ’sinÎ±sinÎ²

\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \cdot \tan \beta}tan(Î±+Î²)=

1âˆ’tanÎ±â‹…tanÎ²

tanÎ±+tanÎ²

â€‹

\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \betasin(Î±âˆ’Î²)=sinÎ±cosÎ²âˆ’cosÎ±sinÎ²

\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \betacos(Î±âˆ’Î²)=cosÎ±cosÎ²+sinÎ±sinÎ²

\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \beta}tan(Î±âˆ’Î²)=

1+tanÎ±â‹…tanÎ²

tanÎ±âˆ’tanÎ²

â€‹

Double angle identities: These identities provide trigonometric formulas for double angle of the triangle. For cosine, there are three formulas which can be derived from each other using Pythagorean identities.

\sin (2\theta ) = 2\sin \theta \cos \thetasin(2Î¸)=2sinÎ¸cosÎ¸

\cos (2\theta ) = {\cos ^2}\theta – {\sin ^2}\theta = 2{\cos ^2}\theta – 1 = 1 – 2{\sin ^2}\thetacos(2Î¸)=cos

2

Î¸âˆ’sin

2

Î¸=2cos

2

Î¸âˆ’1=1âˆ’2sin

2

Î¸

\tan (2\theta ) = \frac{{2\tan \theta }}{{1 – {{\tan }^2}\theta }}tan(2Î¸)=

1âˆ’tan

2

Î¸

2tanÎ¸

â€‹

Second string trigonometric identities

These identities are not so common and only come into consideration when in use. But these identities have much importance in solving the complex equations. Although these identities have lesser number of applications, but when needed can be put directly which will ease the solution.

Supplementary identities: These identities provide the trigonometric formulas for supplementary angles (pi-Î¸) of the triangle.

\sin (\pi – \theta ) = \sin \thetasin(Ï€âˆ’Î¸)=sinÎ¸

\cos (\pi – \theta ) = – \cos \thetacos(Ï€âˆ’Î¸)=âˆ’cosÎ¸

\tan (\pi – \theta ) = – \tan \thetatan(Ï€âˆ’Î¸)=âˆ’tanÎ¸

Half angle identities: In these identities, the negative and positive sign can be taken depending upon the quadrant of angle Î¸/2 i.e., if it is an acute angle then positive value will be taken.

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

2

Î¸

â€‹

=Â±

2

1âˆ’cosÎ¸

â€‹

â€‹

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

2

Î¸

â€‹

=Â±

2

1+cosÎ¸

â€‹

â€‹

\tan \frac{\theta}{2}=\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta}tan

2

Î¸

â€‹

=

1+cosÎ¸

sinÎ¸

â€‹

=

sinÎ¸

1âˆ’cosÎ¸

â€‹

Product sum identities: These identities are useful if one wants to change sine and cosines sum and difference into their products.

\sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}sinÎ±+sinÎ²=2sin

2

Î±+Î²

â€‹

cos

2

Î±âˆ’Î²

â€‹

\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}cosÎ±+cosÎ²=2cos

2

Î±+Î²

â€‹

cos

2

Î±âˆ’Î²

â€‹

\sin \alpha-\sin \beta=2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}sinÎ±âˆ’sinÎ²=2cos

2

Î±+Î²

â€‹

sin

2

Î±âˆ’Î²

â€‹

\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}cosÎ±+cosÎ²=2cos

2

Î±+Î²

â€‹

cos

2

Î±âˆ’Î²

â€‹

Product identities: These identities provide the multiplication of sine and cosine functions with each other or with themselves.

\sin \alpha \cos \beta=\frac{\sin (\alpha+\beta)+\sin (\alpha-\beta)}{2}sinÎ±cosÎ²=

2

sin(Î±+Î²)+sin(Î±âˆ’Î²)

â€‹

\cos \alpha \cos \beta=\frac{\cos (\alpha+\beta)+\cos (\alpha-\beta)}{2}cosÎ±cosÎ²=

2

cos(Î±+Î²)+cos(Î±âˆ’Î²)

â€‹

\sin \alpha \sin \beta=\frac{\cos (\alpha+\beta)-\cos (\alpha-\beta)}{2}sinÎ±sinÎ²=

2

cos(Î±+Î²)âˆ’cos(Î±âˆ’Î²)

â€‹

Triple angle identities: If the angle of the triangle gets tripled in which respect the trigonometric functions are being calculated, then these identities can be used directly for problem solving.

\sin (3 \theta)=3 \sin \theta-4 \sin ^{3} \thetasin(3Î¸)=3sinÎ¸âˆ’4sin

3

Î¸

\cos (3 \theta)=4 \cos ^{3} \theta-\cos \thetacos(3Î¸)=4cos

3

Î¸âˆ’cosÎ¸

\tan (3 \theta)=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta}tan(3Î¸)=

1âˆ’3tan

2

Î¸

3tanÎ¸âˆ’tan

3

Î¸

â€‹

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