List of Trigonometric Identities

There are a lot of identities in the study of Trigonometry, which involves all the trigonometric ratios. These identities are used to solve various problems throughout the academic landscape as well as the real life. Let us learn all the fundamental and advanced trigonometric identities.

Reciprocal Trigonometric Identities

In all trigonometric ratios, there is a reciprocal relation between a pair of ratios, which is given as follows:

• sin θ = 1/cosec θ
• cosec θ = 1/sin θ
   
• cos θ = 1/sec θ 
• sec θ = 1/cos θ
   
• tan θ = 1/cot θ
• cot θ = 1/tan θ

Pythagorean Trigonometric Identities

Pythagorean trigonometric identities are based on the Right-Triangle theorem or Pythagoras theorem, and are as follows:

• sin² θ + cos² θ = 1
• 1 + tan² θ = sec² θ
• cosec² θ = 1 + cot² θ

Read More about Pythagorean Trigonometric Identities.

Trigonometric Ratio Identities

As tan and cot are defined as the ratio of sin and cos, which is given by the following identities:

• tan θ = sin θ/cos θ
• cot θ = cos θ/sin θ

Trigonometric Identities of Opposite Angles

In trigonometry angle measured in the clockwise direction is measured in negative parity and all trigonometric ratios defined for negative parity of angle are defined as follows:

• sin (-θ) = -sin θ
• cos (-θ) = cos θ
• tan (-θ) = -tan θ
• cot (-θ) = -cot θ
• sec (-θ) = sec θ
• cosec (-θ) = -cosec θ

Complementary Angles Identities

Complementary angles are the pair of angles whose measure add up to 90°. Now, the trigonometric identities for complementary angles are as follows:

• sin (90° – θ) = cos θ
• cos (90° – θ) = sin θ
• tan (90° – θ) = cot θ
• cot (90° – θ) = tan θ
• sec (90° – θ) = cosec θ
• cosec (90° – θ) = sec θ

Supplementary Angles Identities

Supplementary angles are the pair of angles whose measure add up to 180°. Now, the trigonometric identities for supplementary angles are:

• sin (180°- θ) = sinθ
• cos (180°- θ) = -cos θ
• cosec (180°- θ) = cosec θ
• sec (180°- θ)= -sec θ
• tan (180°- θ) = -tan θ
• cot (180°- θ) = -cot θ

Periodicity of Trigonometric Function

Trigonometric functions such as sin, cos, tan, cot, sec, and cosec all are periodic in nature and have different periodicity. The following identities for the trigonometric ratio explain their periodicity.

• sin (n × 360° + θ) = sin θ
• sin (2nπ + θ) = sin θ
   
• cos (n × 360° + θ) = cos θ
• cos (2nπ + θ) = cos θ
   
• tan (n × 180° + θ) = tan θ
• tan (nπ + θ) = tan θ
   
• cosec (n × 360° + θ) = cosec θ
• cosec (2nπ + θ) = cosec θ
   
• sec (n × 360° + θ) = sec θ
• sec (2nπ + θ) = sec θ
   
• cot (n × 180° + θ) = cot θ
• cot (nπ + θ) = cot θ

Where, n ∈ Z, (Z = set of all integers)

Note: sin, cos, cosec, and sec have a period of 360° or 2π radians, and for tan and cot period is 180° or π radians.

Sum and Difference Identities

Trigonometric identities for Sum and Difference of angle include the formulas such as sin(A+B), cos(A-B), tan(A+B), etc.

• sin (A+B) = sin A cos B + cos A sin B
• sin (A-B) = sin A cos B – cos A sin B
• cos (A+B) = cos A cos B – sin A sin B
• cos (A-B) = cos A cos B + sin A sin B
• tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
• tan (A-B) = (tan A – tan B)/(1 + tan A tan B) 

Note: Identities for sin (A+B), sin (A-B), cos (A+B), and cos (A-B) are called Ptolemy’s Identities.

Double Angle Identities

Using the trigonometric identities of the sum of angles, we can find a new identity which is called the Double angle Identity. To find these identities we can put A = B in the sum of angle identities. For example,

a  we know, sin (A+B) = sin A cos B + cos A sin B

Substitute A = B = θ on both sides here, and we get:

sin (θ + θ) = sinθ cosθ + cosθ sinθ

• sin 2θ = 2 sinθ cosθ

Similarly,

• cos 2θ = cos²θ – sin ²θ = 2 cos ² θ – 1 = 1 – sin ² θ
• tan 2θ = (2tanθ)/(1 – tan²θ)

Read More about Double Angle Identities.

Half Angle Formulas

Using double-angle formulas, half-angle formulas can be calculated. To calculate half-angle formulas replace θ with θ/2 then,

• [Tex]\sin \frac{\theta}{2}  = \pm \sqrt{\frac{1-\cos \theta}{2}} [/Tex]
• [Tex]\cos \frac{\theta}{2}  = \pm \sqrt{\frac{1+\cos \theta}{2}}  [/Tex]
• [Tex]\tan \frac{\theta}{2}  = \pm\sqrt{\frac{1-\cos \theta}{1+\cos \theta}} =\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta} [/Tex]

Read More about Half Angle Identities.

Some more Half Angle Identities

Other than the above-mentioned identities, there are some more half-angle identities which are as follows:

• [Tex] \sin \theta=\frac{2 \tan \theta / 2}{1+\tan ^2 \theta / 2}  [/Tex]
• [Tex]\cos \theta=\frac{1+\tan ^2 \theta / 2}{1- \tan ^2 \theta / 2}  [/Tex]
• [Tex] \tan \theta = \frac{2 \tan \theta / 2}{1- \tan ^2 \theta / 2} [/Tex]

Product-Sum Identities

The following identities state the relationship between the sum of two trigonometric ratios with the product of two trigonometric ratios.

• [Tex]\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}  [/Tex]
• [Tex]\cos A+\cos B=2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}  [/Tex]
• [Tex]\sin A-\sin B=2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}  [/Tex]
• [Tex]\cos A-\cos B=-2 \sin \frac{A+B}{2} \sin \frac{A-B}{2} [/Tex]

Products Identities

Product Identities are formed when we add two of the sum and difference of angle identities and are as follows:

• [Tex]\sin A \cos B=\frac{\sin (A+B)+\sin (A-B)}{2}  [/Tex]
• [Tex]\cos A \cos B=\frac{\cos (A+B)+\cos (A-B)}{2}  [/Tex]
• [Tex]\sin A \sin B=\frac{\cos (A-B)-\cos (A+B)}{2} [/Tex]

Triple Angle Formulas

Other than double and half angle formulas, there are identities for trigonometric ratios which are defined for triple angle. These identities are as follows:

• [Tex]\sin 3 \theta=3 \sin \theta-4 \sin ^3 \theta  [/Tex]
• [Tex]\cos 3 \theta= 4 \cos^3 \theta-3 \cos \theta  [/Tex]
• [Tex]\cos 3 \theta=\frac{3 \tan \theta-\tan ^3 \theta}{1-3 \tan ^2 \theta}  [/Tex]

Read More about Triple Angle Identities.

Trigonometric Identities are various identities that are used to simplify various complex equations involving trigonometric functions. Trigonometry is a branch of Mathematics that deals with the relationship between the sides and angles of a triangle., These relationships are defined in the form of six ratios which are called trigonometric ratios – sin, cos, tan, cot, sec, and cosec.

In an extended way, the study is also of the angles forming the elements of a triangle. Logically, a discussion of the properties of a triangle; solving a triangle, and physical problems in the area of heights and distances using the properties of a triangle – all constitute a part of the study. It also provides a method of solution to trigonometric equations.