Trigonometric Identities Class 10 formulas proofs and solved examples
The concept of Trigonometric Identities Class 10 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Trigonometric Identities Class 10?
Trigonometric identities class 10 are special mathematical equations that show consistent relationships between the trigonometric ratios: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). These identities are always true for defined values of an angle and help simplify complex trigonometric expressions. You’ll find this concept applied in areas such as trigonometric ratios, problem-solving in trigonometry, and applications of trigonometry in geometry and real life.
Key Formula for Trigonometric Identities Class 10
Here are the three core trigonometric identities that you need to know for class 10:
| Identity | Formula | True For |
|---|---|---|
| Pythagorean (Sin & Cos) | sin2θ + cos2θ = 1 | 0° ≤ θ ≤ 90° |
| Tan & Sec | 1 + tan2θ = sec2θ | 0° ≤ θ < 90° |
| Cot & Cosec | 1 + cot2θ = cosec2θ | 0° < θ ≤ 90° |
Other important formulas related to trigonometric identities are also used to express one trigonometric ratio in terms of others (for example, tan θ = sin θ / cos θ and sec θ = 1 / cos θ).
Cross-Disciplinary Usage
Trigonometric identities class 10 are not only useful in Maths but also play an important role in Physics, engineering, and daily logical reasoning. Students preparing for competitive exams like JEE, NEET, and Olympiads see these identities used in questions related to waves, oscillations, and angle measurements. Vedantu lessons often extend these connections for deeper understanding.
Step-by-Step Illustration
Let’s prove that sin2A + cos2A = 1 for any angle A (0° ≤ A ≤ 90°):
1. Consider a right-angled triangle with sides: hypotenuse = AB, base = AC, perpendicular = BC.2. By Pythagoras’ theorem: AB2 = AC2 + BC2
3. Divide both sides by AB2:
4. (AB2)/(AB2) = (AC2)/(AB2) + (BC2)/(AB2)
5. 1 = (AC/AB)2 + (BC/AB)2
6. From trigonometric definitions: cos A = AC/AB and sin A = BC/AB
7. Substituting: 1 = (cos A)2 + (sin A)2
8. Or, sin2A + cos2A = 1
This identity helps to simplify many board exam problems quickly.
Speed Trick or Vedic Shortcut
Here’s a quick trick: If a question asks you to find 1 − sin2A, just recall the identity sin2A + cos2A = 1 so that:
1. 1 − sin2A = cos2A
This shortcut makes trigonometric simplifications in MCQ questions speedy and reduces calculation mistakes, especially during time-pressured board exams.
Try These Yourself
- Simplify the expression: sin2θ + cos2θ + tan2θ − sec2θ.
- If cos A = 3/5, find sin A using an identity.
- Prove that sec2P − tan2P = 1.
- Express cot θ in terms of cosec θ.
Frequent Errors and Misunderstandings
- Forgetting the squares in sin2θ or cos2θ, leading to wrong simplifications.
- Mixing up identities: e.g., using tan2θ + cos2θ = 1 (which is NOT correct).
- Missing domain restrictions (e.g., tan 90° is undefined).
Relation to Other Concepts
The idea of trigonometric identities class 10 connects closely with trigonometric ratios, complementary angles, and applications such as height and distance problems. Mastering these helps in understanding the use of formulas and angle transformations in both boards and higher classes.
Classroom Tip
A quick way to remember the core identities is the “Magic 1” trick: Every identity has 1 on one side (sin2θ + cos2θ = 1, 1 + tan2θ = sec2θ, 1 + cot2θ = cosec2θ). Vedantu teachers often draw triangle diagrams and identity charts to help students visualize and memorize quickly.
We explored trigonometric identities class 10—from definition, formula, stepwise sample, common doubts, and links to other maths topics. Continue practicing with Vedantu to become confident in solving trigonometry questions and excel in your exams.
FAQs on Trigonometric Identities for Class 10 Maths
1. What are trigonometric identities in Class 10?
Trigonometric identities are equations involving trigonometric ratios that are true for all values of the angle where they are defined. In Class 10, these identities mainly relate sin θ, cos θ, and tan θ. They are used to simplify expressions and prove other identities in trigonometry. The most important identities are:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
2. What is the fundamental identity in trigonometry?
The fundamental trigonometric identity is sin²θ + cos²θ = 1. This identity is derived from the Pythagoras theorem in a right-angled triangle. By dividing the Pythagorean relation by hypotenuse², we obtain this identity. It is called fundamental because other identities like 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are derived from it.
3. How do you prove trigonometric identities in Class 10?
To prove a trigonometric identity, simplify one side of the equation until it becomes equal to the other side. Follow these steps:
- Start with the more complicated side (LHS or RHS).
- Use known identities like sin²θ + cos²θ = 1.
- Convert all ratios into sin θ and cos θ if needed.
- Simplify step by step without changing both sides together.
4. What is the formula for tan θ in terms of sin θ and cos θ?
The formula for tangent in terms of sine and cosine is tan θ = sin θ / cos θ. This formula comes from the basic trigonometric ratios in a right-angled triangle:
- sin θ = Perpendicular / Hypotenuse
- cos θ = Base / Hypotenuse
5. What are the three main trigonometric identities for Class 10?
The three main trigonometric identities in Class 10 are:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
6. Can you give an example of proving a trigonometric identity?
Yes, for example, prove that (1 − tan²θ)/(1 + tan²θ) = cos 2θ is not in Class 10, so instead consider proving 1 + tan²θ = sec²θ. Start with LHS:
- 1 + tan²θ
- = 1 + (sin²θ / cos²θ)
- = (cos²θ + sin²θ) / cos²θ
- = 1 / cos²θ
- = sec²θ
7. Why is sin²θ + cos²θ equal to 1?
The identity sin²θ + cos²θ = 1 is true because it is derived from the Pythagoras theorem in a right triangle. If a right triangle has perpendicular = P, base = B, and hypotenuse = H, then:
- P² + B² = H²
- Dividing by H² gives (P/H)² + (B/H)² = 1
- Since P/H = sin θ and B/H = cos θ, we get sin²θ + cos²θ = 1
8. What are the reciprocal trigonometric identities?
Reciprocal trigonometric identities show the inverse relationship between ratios. They are:
- sec θ = 1 / cos θ
- cosec θ = 1 / sin θ
- cot θ = 1 / tan θ
9. What are common mistakes students make in trigonometric identities?
Common mistakes in trigonometric identities include incorrect substitution and algebraic errors. Students often:
- Forget that tan θ = sin θ / cos θ
- Apply identities to both sides at once
- Make calculation mistakes in squaring terms like sin²θ
- Ignore domain restrictions where ratios are undefined
10. How are trigonometric identities useful in solving problems?
Trigonometric identities are useful because they help simplify complex expressions and prove equations easily. In Class 10 mathematics, they are used to:
- Simplify expressions involving sin, cos, and tan
- Prove LHS = RHS in identity problems
- Transform one trigonometric ratio into another
- Solve basic trigonometric equations
