Trigonometric Identities Class 10
Trigonometric Identities are the rules that are followed by the Trigonometric Ratios. Trigonometric Identities and ratios are the fundamentals of trigonometry. Trigonometry is used in various fields for different calculations. The trigonometric identities class 10 gives the connection between the different trigonometric ratios. This article covers trigonometric identities class 10 in addition to their proofs.
This article will be extremely helpful for class 10 students for their exams. Also, we will solve some examples of trigonometric identities Class 10. Let’s discuss the topic of Trigonometric Identities Class 10 in depth.
Table of Content
- What are Trigonometric Identities?
- What are Trigonometric Ratios?
- List of Trigonometric Identities Class 10
- Proof of Trigonometric Identities Class 10
- Solved Examples
What are Trigonometric Identities?
The identities which provide the relation between the different trigonometric ratios are called Trigonometric Identities. In class 10, the trigonometric identities relate all the trigonometric ratios with each other. The trigonometric identities class 10 includes three trigonometric identities. Below we will study the trigonometric identities of class 10.
What are Trigonometric Ratios?
There are six basic trigonometric ratios namely, sin, cos, tan, sec, cosec and cot. The formulas of these basic trigonometric ratios are listed below:
- Sin θ = Perpendicular / Hypotenuse
- Cos θ = Base / Hypotenuse
- Tan θ = Sin θ / Cos θ = Perpendicular / Base
- Cosec θ = 1 / Sin θ = Hypotenuse / Perpendicular
- Sec θ = 1 / Cos θ = Hypotenuse / Base
- Cot θ = 1 / Tan θ = Base / Perpendicular
List of All Trigonometric Identities Class 10
The three basic trigonometric identities are listed below:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
Some Other Trigonometric Identities Class 10
Other then Pythagorean identities there are some more identities discussed in class 10. These identities are:
Reciprocal Identities
- cosec θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
Quotient Identities
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
Learn more about Trigonometrtic Identities
Proof of Trigonometric Identities Class 10
In this section we will prove trigonometric identities class 10 that are mentioned above.
Consider a right-angled triangle PQR.
By Pythagoras theorem
PR2 = PQ2 + QR2 – – -(1)
Proof of sin2θ + cos2θ = 1
Dividing equation (1) by PR2
1 = PQ2/PR2 + QR2/PR2
⇒ 1 = (PQ/PR)2 + (QR/PR)2 – – -(2)
In right-angled triangle PQR
PQ/PR = sin θ
and QR/PR = cos θ
Put value of PQ/PR and QR/PR in equation (2), we get
sin2θ + cos2θ = 1
Hence proved.
Proof of 1 + tan2θ = sec2θ
Dividing equation (1) by QR2
PR2/QR2= PQ2/QR2 + 1
⇒ (PR/QR)2 = (PQ/QR)2 + 1 – – -(3)
In right-angled triangle PQR
PR/QR = sec θ
and PQ/QR = tan θ
Put value of PR/QR and PQ/QR in equation (3), we get
1 + tan2θ = sec2θ
Hence proved.
Proof of 1 + cot2θ = cosec2θ
Dividing equation (1) by PQ2
(PR2/PQ2) = 1 + (QR2/PQ2)
(PR/PQ)2 = 1 + (QR/PQ)2 – – – (4)
In right-angled triangle PQR
PR/PQ = cosec θ
and QR/PQ = cot θ
Put value of PR/PQ and QR/PQ in equation (4), we get
1 + cot2θ = cosec2θ
Hence proved.
Trigonmetric Identities Class 10 Table
Identities Name |
Identities |
---|---|
Pythagorean Identities | sin2θ + cos2θ = 1 |
1 + tan2θ = sec2θ | |
1 + cot2θ = cosec2θ | |
Reciprocal Identities | cosec θ = 1/sin θ |
sec θ = 1/cos θ | |
cot θ = 1/tan θ | |
Quotient Identities | tan θ = sin θ/cos θ |
cot θ = cos θ/sin θ |
Also, Check
Solved Examples on Trigonometric Identities Class 10
Example 1: Prove that: (1 + cot2A) sin2A = 1
Solution:
LHS = (1 + cot2A) sin2A
Using trigonometric identity
1 + cot2θ = cosec2θ
⇒ LHS = cosec2A sin2A
⇒ LHS = (1 / sin2A) sin2A [cosec A = 1 / sin A]
⇒ LHS = 1
Thus, LHS = RHS
Hence Proved
Example 2: Prove that: tan2B cos2B = 1 – cos2B
Solution:
LHS = tan2B cos2B
Using trigonometric identity
⇒ 1 + tan2θ = sec2θ
⇒ tan2θ = sec2θ – 1
⇒ LHS = (sec2B – 1) cos2B
⇒ LHS = sec2Bcos2B – cos2B
⇒ LHS = [(1/ cos2B)cos2B] – cos2B
⇒ LHS = 1 – cos2B
Thus, LHS = RHS
Hence Proved
Example 3: Prove that: tan θ + (1/tan θ) = sec θ cosec θ
Solution:
LHS = tan θ + (1/tan θ)
⇒ LHS = (tan2θ + 1)/tan θ
Using trigonometric identity
1 + tan2θ = sec2θ
⇒ LHS = sec2θ/tanθ
⇒ LHS = (1 / cos2θ) × (cos θ / sinθ )
⇒ LHS = (1 / cos θ) × (1 / sin θ)
⇒ LHS = sec θ cosec θ [sec θ = 1/cos θ and cosec θ = 1/sin θ]
Thus, LHS = RHS
Hence Proved
Example 4: Prove that: √[(1 – cos θ)/(1 + cos θ)] = cosec θ – cot θ
Solution:
LHS = √[(1 – cos θ)/(1 + cos θ)]
⇒ LHS = √[{(1 – cos θ)/(1 + cos θ)} × {(1 – cos θ) / (1 – cos θ)}]
⇒ LHS = √[(1 – cos θ)2/(1 – cos2θ)]
Using trigonometric identity
sin2θ + cos2θ = 1
⇒ LHS = √[(1 – cosθ)2 / sin2θ]
⇒ LHS = (1 – cos θ) / sin θ
⇒ LHS = (1 / sin θ) – (cos θ / sin θ)
⇒ LHS = cosec θ – cot θ [cotθ = cosθ / sinθ and cosecθ = 1/sinθ]
Thus, LHS = RHS
Hence Proved
Example 5: Prove that: sin θ / (1 – cos θ) = cosec θ + cot θ
Solution:
LHS = sin θ/(1 – cos θ)
⇒ LHS = [sin θ/(1 – cos θ)] × [(1 + cos θ)/(1 + cos θ)}]
⇒ LHS = [sin θ (1 + cos θ)] / (1 – cos2θ)
Using trigonometric identity
sin2θ + cos2θ = 1
⇒ LHS = [sin θ (1 + cos θ)] / sin2θ
⇒ LHS = (1 + cos θ)/sin θ
⇒ LHS = (1 / sin θ) + (cos θ/sinθ)
⇒ LHS = cosec θ + cot θ [cot θ = cos θ/sin θ and cosec θ = 1/sin θ]
Thus, LHS = RHS
Hence Proved
Example 6: Prove that: [(1 + cot2θ)tan θ]/sec2θ = cot θ
Solution:
LHS = [(1 + cot2θ)tan θ]/sec2θ
Using trigonometric identity
1 + cot2θ = cosec2θ
⇒ LHS = [cosec2θ tan θ]/sec2θ
⇒ LHS = (1/sin2θ) × (sin θ/cos θ) × (cos2θ/1)
⇒ LHS = (cos θ/ sin θ)
⇒ LHS = cot θ [cot θ = cos θ/sinθ]
Thus, LHS = RHS
Hence Proved
Practice Questions on Trigonometric Identities Class 10
Q1: Prove that: cosecθ√[(1 – cos2θ) = 1
Q2: Prove that: (sec2x – 1)(cosec2x – 1) = 1
Q3: Prove that: sin2C + [1 / (1 + tan2C)] = 1
Q4: Prove that: sec A (1 – sin A) (sec A + tan A) = 1
Q5: Prove that: (1 – sin θ) / (1 + sin θ) = (sec θ – tanθ)2
Q6: Prove that: (cosec θ + sin θ)(cosec θ – sin θ) = cot2θ + cos2θ
Trigonometric Identities Class 10 – FAQs
What are Trigonometric Identities?
The trigonometric identities are the identities which gives the connection between the trigonometric identities.
List the Three basic Trigonometric Identities in Class 10.
The three basic trigonometric identities are:
- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
What are the Six Trigonometric Ratios?
The six basic trigonometric ratios are: sin, cos, tan, cosec, sec and cot.
How to Prove Trigonometric Identities Class 10?
To prove trigonometric identities class 10 we use Pythagoras theorem. The detailed proof has been discussed in the article above.
Is Trigonometric Identities Class 10 Important?
Yes, Trigonometric Identities Class 10 is extremely important for board exams as well as for building strong foundation in Trigonometry
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