Pythagorean Identities in trigonometry are derived from Pythagorean Theorem. They are also called Pythagorean Trigonometric Identities. One of the basic Pythagorean identity is sin²θ + cos²θ = 1. Let's learn more about, Pythagorean identities in trigonometry including their proof and solved examples, as well.
What are Pythagorean Identities?
Pythagorean identities are important identities in trigonometry that are based on the Pythagoras theorem. It relates the square of one trigonometric ratio with the other. It can be used to solve complex trigonometric problems easily and also used to prove various other trigonometric identities.
Pythagorean Trig Identities
Pythagorean identities are the identities which gives us the relation between the square of one trigonometric ratio to square of another trigonometric ratio.
These identities are called Pythagorean identities because they follow Pythagoras theorem. In trigonometry, we have three basic Pythagorean identities which are:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
Pythagoras Theorem
Pythagoras theorem states that in a right-angled triangle, the square of hypotenuse is equal to the sum of squares of perpendicular and base. The formula for Pythagoras theorem is given by:
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
In the above figure is of right-angled triangle ABC with perpendicular AB, base BC and hypotenuse AC. So, by the Pythagoras theorem: AC2 = AB2 + BC2
List of Pythagorean Identities
As we know the three basic Pythagorean identities in trigonometry are:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
Variation of Pythagorean Identities
However, from these identities we can derive the other six identities as well, which are given by:
- sin2θ + cos2θ = 1
- 1 - cos2θ = sin2θ
- 1 - sin2θ = cos2θ
- 1 + tan2θ = sec2θ
- sec2θ - tan2θ = 1
- tan2θ = sec2θ - 1
- 1 + cot2θ = cosec2θ
- cosec2θ - cot2θ = 1
- cot2θ = cosec2θ - 1
Pythagorean Trig Identities Derivation
Pythagorean identities can be easily proved by applying Pythagorean theorem in a right-angled triangle. Let's discuss the proof of each identity in detail as follows:
Derivation of sin2θ + cos2θ = 1
Consider a right-angled triangle ABC with perpendicular AB, base BC and hypotenuse AC.
By the Pythagoras theorem
AC2 = AB2 + BC2
Dividing both side by AC2
[AC2 / AC2] = [AB2 / AC2] + [BC2 / AC2]
⇒ 1 = [AB / AC]2 + [BC / AC]2 . . . (a)
From the above triangle
[AB / AC] = sinθ, and
[BC / AC] = cosθ
Putting these values in equation a, we get
1 = sin2θ + cos2θ, or
⇒ sin2θ + cos2θ = 1
Hence Proved.
Derivation of 1 + tan2θ = sec2θ
Consider a right-angled triangle ABC with perpendicular AB, base BC and hypotenuse AC.
By the Pythagoras theorem
AC2 = AB2 + BC2
Dividing both side by BC2
[AC2 / BC2] = [AB2 / BC2] + [BC2 / BC2]
⇒ AC / BC]2 = [AB / BC]2 + 1 . . . (b)
From the above triangle, we have
[AC / BC] = sec θ, and
[AB / BC] = tan θ
Putting these values in equation b, we get
sec2θ = tan2θ + 1, or
⇒ sec2θ - tan2θ = 1
Hence Proved
Derivation of 1 + cot2θ = cosec2θ
Consider a right-angled triangle ABC with perpendicular AB, base BC and hypotenuse AC.
By the Pythagoras theorem, we have
AC2 = AB2 + BC2
Dividing both side by AB2, we get
[AC2 / AB2] = [AB2 / AB2] + [BC2 / AB2]
⇒ [AC / AB]2 = 1 + [BC / AB]2 . . . (c)
From the above triangle, we know
[AC / AB] = cosec θ, and
[BC/ AB] = cot θ
Putting these values in equation c we get,
⇒ cosec2θ = 1 + cot2θ, or
⇒ cosec2θ - cot2θ = 1
Hence Proved
Applications of Pythagorean Identities
Some applications of Pythagorean identities in trigonometry are:
- Pythagorean identities can be used to derive other trigonometric identities.
- It can also be used in calculation of heights and distances.
- It can be used to simplify trigonometric problems.
Other Trigonometric Identities
Some of the other identities include:
Conclusion (Pythagorean Trigonometric Identities)
From the above discussion we can conclude that we have three main Pythagorean identities in trigonometry. The three Pythagorean identities are
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
These Pythagorean identities are proved using Pythagoras theorem. The Pythagoras theorem says that in a right-angled triangle the square of the longest side is equal to the sum of squares of other two sides. The Pythagorean identities are useful in the derivation and simplification of other trigonometric identities and problems.
Read More,
Pythagorean Identities Examples
Example 1: If the angle y is in second quadrant and cos y = 5/13. Find the value of sin y.
Solution:
By Pythagorean identity
sin2y + cos2y = 1
⇒ sin2y + (5/13)2= 1
⇒ sin2y = 1 - (25/169)
⇒ sin2y = 144 / 169
⇒ sin y = ± 12/13
Since, sin is positive in second quadrant so we will take positive value i.e., sin y = 12/13.
Example 2: Evaluate [1/ (1 - cos x)] + [1/ (1 + cos x)]
Solution:
Given,
- [1/ (1 - cos x)] + [1/ (1 + cos x)]
= [(1 + cosx) + (1 - cos x)] / [(1 + cosx) (1 - cos x)]
= 2 / (12- cos2x)
= 2 / (1 - cos2x)
By Pythagorean identity, sin2y + cos2y = 1 or sin2y = 1 - cos2y
⇒ [1/ (1 - cos x)] + [1/ (1 + cos x)] = 2 / sin2x
⇒ [1/ (1 - cos x)] + [1/ (1 + cos x)] = 2 cosec2x
Example 3: Simplify (tanθ + secθ)(tanθ - secθ)
Solution:
Given,
- (tan θ + sec θ)(tan θ - sec θ)
= (tan2 θ - sec2 θ) [ As (a-b)(a+b) = a2 - b2]
By Pythagorean identity, we have
sec2 θ - tan2 θ = 1
(tan θ + sec θ)(tan θ - sec θ) = -(sec2 θ - tan2 θ) = -1
Hence, (tan θ + sec θ)(tan θ - sec θ) = -1
Example 4: Evaluate (cosec z + cot z) [(1 - cos z)/sin z]
Solution:
Given,
- (cosec z + cot z) [(1 - cos z)/sin z]
Simplifying,
(cosec z + cot z) [(1/sin z) - (cos z/sin z)]
= (cosec z + cot z) (cosec z - cot z) [As 1/ sin z = cosec z and cos z/sinz = cot z]
= (cosec2z - cot2z) [ As (a-b)(a+b) = a2 - b2]
By Pythagorean Identity we have, cosec2θ - cot2θ = 1
Hence, (cosec z + cot z) [(1 - cos z)/sin z] = 1
Practice Questions on Pythagorean Trig Identities
Q1: If cot A = 4/3 then, find the value of cosec A.
Q2: Evaluate [1/ (1 - cosec x)] [1/ (1 + cosec x)].
Q3: Simplify (cosecθ + cotθ)(cosecθ - cotθ).
Q4: Prove that: sin4x - cos4x = sin2x - cos2x.
FAQs on Pythagorean Identities
What are Pythagorean Identities?
The identities in trigonometry that are based on Pythagoras theorem are called as Pythagorean Identities.
What is Pythagoras Theorem?
Pythagoras theorem states that in a right-angled triangle the square of hypotenuse is equal to the sum of squares of other two sides.
What are the 3 Pythagorean Identities?
3 Pythagorean Identities are:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
How to Prove Pythagorean Identities in Trigonometry?
To prove Pythagorean identities in trigonometry we use Pythagoras theorem.
What is Pythagorean Triple Rule?
Pythagorean triple rule is in a right-angled triangle (hypotenuse)2 = (Perpendicular)2 + (Base)2.
What are two Quotient Identities?
The 2 quotient identities are:
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
List all Pythagorean Identities.
All 9 Pythagorean identities are:
| Pythagorean Identity | Alternative Identites |
|---|
| sin2θ + cos2θ = 1 | 1 - sin2θ = cos2 θ OR 1 - cos2θ = sin2θ |
| sec2θ - tan2θ = 1 | 1 + tan2θ = sec2θ OR sec2θ - 1 = tan2θ |
| cosec2θ - cot2θ = 1 | 1 + cot2θ = cosec2θ OR cosec2θ - 1 = cot2θ |
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