Trigonometric Identities

Trigonometric Identities are various identities that are used to simplify various complex equations involving trigonometric functions. Trigonometry is a branch of Mathematics, which relates to the study of angles, measurement of angles, and units of measurement. It also concerns itself with the six ratios for a given angle and the relations satisfied by these ratios. 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.

Trigonometric Identity

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.

List of Trigonometric Identities

Various identities in trigonometry are used to solve many problems. Let us learn all the fundamental trigonometric identities.

Reciprocal Trigonometric Identities

Various reciprocal trigonometric identities are:

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

Pythagorean Trigonometric Identities

Pythagorean trigonometric identities are based on the Right-Triangle theorem or Pythagoras theorem.

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

Trigonometric Ratio Identities

Trigonometric ratio identities are:

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

Trigonometric Identities of Opposite Angles

List of opposite angle trigonometric identities are:

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

Trigonometric Identities of Complementary Angles

Complementary angles are the angles if their sum is equal to 90°. Now, the trigonometric identities for complementary angles are:

• 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

Trigonometric Identities of Supplementary Angles

Supplementary angles are the angles if their sum is equal to 90°. 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

Trigonometric Identites for Sum and Difference

Sum and Difference of trigonometric identities 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) 

Trigonometric Identites of Double angle

Double angle trigonometric identities are calculated with the help of sum and difference formulas. For example,

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)

Trigonometric identites of Half Angle

Using double angle formulas, half-angle formulas can be calculated

To calculate half angle formulas replace θ by θ/2 then,

• sin (θ/2) = ±√[(1 – cosθ)/2]
• cos (θ/2) = ±√(1 + cosθ)/2
• tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

Proof of the Trigonometric Identities

For any acute angle θ, prove that 

(i) tanθ = sinθ/cosθ

(ii) cotθ = cosθ/sinθ

(iii) tanθ . cotθ = 1

(iv) sin²θ + cos²θ = 1

(v) 1 + tan²θ = sec²θ

(vi) 1 + cot²θ = cosec²θ

Proof:

Consider a right-angled △ABC (fig. 1) in which ∠B = 90°

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

Then, 

(i) tanθ = y/x = (y/r) / (x/r) 

∴ tanθ = sinθ/cosθ 

(ii) cotθ = x/y = (x/r) / (y/r)

∴ cotθ = cosθ/sinθ 

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

tanθ . cotθ = 1 

Then, by Pythagoras’ theorem, we have 

x² + y² = r². 

Now, 

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

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

sin²θ + cos²θ = 1

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

(r/x)² = sec²θ 

∴ 1 + tan²θ = sec²θ.

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

(r²/y²) = cosec²θ

∴ 1 + cot²θ = cosec²θ

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.

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

Solution:

We have:

LHS = (1 + tan²θ)cos²θ

= sec²θ . cos²θ

= (1/cos²θ) . cos²θ

= 1 = RHS.

∴ LHS=RHS.

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

Solution:

We have: 

LHS = (cosec²θ – 1) tan²θ 

= (1 + cot²θ – 1) tan²θ  

= cot²θ . tan²θ  

= (1/tan²θ) . tan²θ

= 1 = RHS.

∴ LHS=RHS.

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

Solution:

We have:

LHS = (sec⁴θ – sec²θ)

       = sec²θ(sec²θ – 1)

       = (1 + tan²θ) (1 + tan²θ – 1)

       = (1 + tan²θ) tan²θ

       = (tan²θ + tan⁴θ)

       = RHS      

∴ LHS = RHS.

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

Solution:

We have:

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

       = √(tan²θ + cot²θ + 2)

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

       = √(tanθ + cotθ)²

= tanθ + cotθ = RHS

∴ LHS = RHS

FAQs on Trigonometric Identities

Question 1: Write the three main trigonometric functions.

Answer:

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

Question 2: What is the Pythagoras Theorem?

Answer:

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

Question 3: Write the uses of Trigonometric Identities.

Answer:

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

Question 4: Write Eight Fundamental Trigonometric Identities.

Answer:

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