Trigonometric 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.

What are Trigonometric Identities?

An equation involving trigonometric ratios of an angle is called trigonometric Identity if it is true for all values of the angle. These are useful whenever trigonometric functions are involved in an expression or an equation. The six basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side.

Trigonometric Identities

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

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)

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]

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]

Proof of the Trigonometric Identities

For any acute angle θ, prove that 

1. tanθ = sinθ/cosθ
2. cotθ = cosθ/sinθ
3. tanθ . cotθ = 1
4. sin²θ + cos²θ = 1
5. 1 + tan²θ = sec²θ
6. 1 + cot²θ = cosec²θ

Proof:

Consider a right-angled △ABC in which ∠B = 90°

Let AB = x units, BC = y units and AC = r units. 

Then, 

(1) tanθ = P/B = y/x = (y/r) / (x/r) 

∴ tanθ = sinθ/cosθ 

(2) cotθ = B/P = x/y = (x/r) / (y/r)

∴ cotθ = cosθ/sinθ 

(3) tanθ . cotθ  = (sinθ/cosθ) . (cosθ/sinθ) 

tanθ . cotθ = 1 

Then, by Pythagoras’ theorem, we have 

x² + y² = r². 

Now, 

(4) sin²θ + cos²θ  = (y/r)² + (x/r)² = ( y²/r² + x²/r²)

                              = (x² + y²)/r² = r²/r² = 1 [x²+ y² = r²]

sin²θ + cos²θ = 1

(5) 1 + tan²θ = 1 + (y/x)² = 1 + y²/x² = (y² + x²)/x² = r²/x² [x² + y² = r²]

(r/x)² = sec²θ 

∴ 1 + tan²θ = sec²θ.

(6) 1 + cot²θ = 1 + (x/y)² = 1 + x²/y² = (x² + y²)/y² = r²/y² [x² + y² = r²]

(r²/y²) = cosec²θ

∴ 1 + cot²θ = cosec²θ

Relation between Angles and Sides of Triangle

Three rules which related the sides of triangles to the interior angles of triangles are,

• Sine Rule
• Cosine Rule
• Tangent Rule

If a triangle ABC with sides a, b, and c which are sides opposites to the ∠A, ∠B, and ∠C respectively, then

Sine Rule

Sine rule states the relationship between sides and angles of the triangle which is the ratio of side and sine of angle opposite to the side always remains the same for all the angles and sides of the triangle and is given as follows:

[Tex]\bold{\frac{\sin \angle A}{a}= \frac{\sin \angle B}{b} = \frac{\sin \angle C}{c} = k} [/Tex]

Cosine Rule 

Cosine Rule involves all the sides, and one interior angle of the triangle is given as follows:

[Tex]\bold{\cos \angle A = \frac{b^2+c^2 – a^2}{2bc}} [/Tex]

OR

[Tex]\bold{\cos \angle B = \frac{a^2+c^2 – b^2}{2ac}} [/Tex]

OR

[Tex]\bold{\cos \angle C = \frac{a^2+b^2 – c^2}{2ab}} [/Tex]

Tangent Rule

• Tangent Rule also states the relationship between the sides and interior angle of a triangle, using the tan trigonometric ratio, which is as follows:

• [Tex]\bold{\frac{a-b}{a+b}=\frac{\tan \left(\frac{A-B}{2}\right)}{\tan \left(\frac{A+B}{2}\right)}} [/Tex]
• [Tex]\bold{\frac{b-c}{b+c}=\frac{\tan \left(\frac{B-C}{2}\right)}{\tan \left(\frac{B+C}{2}\right)}} [/Tex]
• [Tex]\bold{\frac{c-a}{c+a}=\frac{\tan \left(\frac{C-A}{2}\right)}{\tan \left(\frac{C+A}{2}\right)}} [/Tex]

Solved Example on Trigonometric Identities

Example 1: Prove that (1 – sin²θ) sec²θ = 1 

Solution:

We have: 

LHS = (1 – sin²θ) sec²θ

= cos²θ . sec²θ 

= cos²θ . (1/cos²θ)

=1 

= RHS. 

∴ LHS = RHS. [Hence Proved]

Example 2: Prove that (1 + tan²θ) cos²θ = 1  

Solution:

We have:

LHS = (1 + tan²θ)cos²θ

⇒ LHS = sec²θ . cos²θ

⇒ LHS = (1/cos²θ) . cos²θ

⇒ LHS = 1 = RHS.

∴ LHS=RHS. [Hence Proved]

Example 3: Prove that (cosec²θ – 1) tan²θ = 1 

Solution:

We have: 

LHS = (cosec²θ – 1) tan²θ 

⇒ LHS = (1 + cot²θ – 1) tan²θ  

⇒ LHS = cot²θ . tan²θ  

⇒ LHS = (1/tan²θ) . tan²θ

⇒ LHS = 1 = RHS.

∴ LHS=RHS. [Hence Proved]

Example 4: Prove that (sec⁴θ – sec²θ) = (tan²θ + tan⁴θ)

Solution:

We have:

LHS = (sec⁴θ – sec²θ)

⇒ LHS = sec²θ(sec²θ – 1)

⇒ LHS = (1 + tan²θ) (1 + tan²θ – 1)

⇒ LHS = (1 + tan²θ) tan²θ

⇒ LHS = (tan²θ + tan⁴θ) = RHS      

∴ LHS = RHS. [Hence Proved]

Example 5: Prove that √(sec²θ + cosec²θ) = (tanθ + cotθ) 

Solution:

We have:

LHS = √(sec²θ + cosec²θ ) = √((1 + tan²θ) + (1 + cot²θ))

⇒ LHS = √(tan²θ + cot²θ + 2)

⇒ LHS = √(tan²θ + cot²θ + 2tanθ.cotθ )         (tanθ . cotθ = 1)

⇒ LHS = √(tanθ + cotθ)²

⇒ LHS = tanθ + cotθ = RHS

∴ LHS = RHS [Hence Proved]

FAQs on Trigonometric Identities

1. What is Trigonometric Identity?

Trigonometric Identity is an equation that relates different trigonometric functions such as sin, cos, tan, cot, sec, and cosec.

2. How to Prove Trigonometric Identities?

There are various methods for proving trigonometric identities, one of such method is using the 6 main trigonometric known identities to rewrite an expression in a different form. Like any other proof we work with one side to come to an expression identical to the other side of the equation.

3. How Many Trigonometric Identities are there?

There are a lot of trigonometric identities, as any identity can be with some variation is still identity as well. Therefore we can’t say exactly how many identities are there.

4. How to remember all the Trigonometric Identities?

The easiest method to remember all the identities is to practice problems related to the identity. Each time you solve a problem using some identity you revise that identity and eventually it will become second nature to you.

5. Write the Three Main Trigonometric Functions.

Three main functions used in trigonometry are Sine, Cosine, and Tangent.
sin θ = Perpendicular/ Hypotenuse
cos θ = Base/Hypotenuse
tan θ = Perpendicular/Base

6. What is the Pythagoras Theorem?

Pythagoras Theorem states in a right-angle triangle with sides as Hypotenuse(H), Perpendicular(P), and Base(B) the relation between them is given by,

(H)² = (P)² + (B)²

7. Write the uses of Trigonometric Identities.

Trigonometric identities are used for solving various problems involving complex trigonometric functions. They are used to calculate wave equations, equation of Harmonic Oscillator, solving Geometrical Questions and other problems.

8. Write Eight Fundamental Trigonometric Identities.

Eight fundamental identities in trigonometry are:

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• sin²θ + cos²θ = 1
• tanθ = sinθ/cos θ
• 1+ tan²θ = sec²θ
• cot θ = cosθ/sinθ
• 1+ cot²θ = cosec²θ