Trigonometric Identities Formula List for Class 10 Students
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: Complete Guide
1. What is the fundamental concept of a trigonometric identity in Class 10 Maths?
A trigonometric identity is an equation involving trigonometric ratios (like sin, cos, tan) that holds true for all possible values of the angle(s) involved. Unlike a regular equation, which is only true for specific values, an identity represents a fundamental and universal relationship between the trigonometric ratios.
2. What are the three main Pythagorean trigonometric identities for Class 10?
As per the CBSE 2025-26 syllabus, students focus on three fundamental Pythagorean identities, which are crucial for solving problems:
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Understanding the first identity is key, as the other two can be derived from it.
3. Why is the identity sin²θ + cos²θ always equal to 1?
This core identity is a direct consequence of the Pythagorean theorem (a² + b² = c²) applied to a right-angled triangle. If you consider a triangle with hypotenuse 'c', opposite side 'a', and adjacent side 'b' with respect to angle θ, we know sinθ = a/c and cosθ = b/c. Substituting these into the identity gives (a/c)² + (b/c)² = (a² + b²)/c². Since a² + b² = c², the expression becomes c²/c², which equals 1. This shows it's a geometric truth, not just an algebraic rule.
4. What is the key difference between a trigonometric identity and a trigonometric equation?
The primary difference lies in their validity and purpose:
An identity is true for all valid values of the angle. For example, tanθ = sinθ/cosθ is always true. The goal is to 'prove' an identity.
An equation is true only for specific values of the angle. For example, 2sinθ = 1 is only true for θ = 30°, 150°, etc. The goal is to 'solve' an equation to find these specific values.
5. What is the general method for proving a trigonometric identity?
To prove a trigonometric identity, you must show that the Left Hand Side (LHS) is algebraically equivalent to the Right Hand Side (RHS). Common strategies include:
- Starting with the more complex side and simplifying it step-by-step.
- Converting all trigonometric ratios into their sine and cosine forms.
- Using the fundamental identities like sin²θ + cos²θ = 1 for substitution.
- Applying algebraic techniques such as factoring, expansion, or finding a common denominator.
You work on one side until it matches the other, or simplify both sides to a common expression.
6. What is the importance of trigonometric identities in simplifying expressions?
The main importance of trigonometric identities is their ability to transform and simplify complex trigonometric expressions. By substituting parts of an expression with an equivalent but simpler form using an identity, we can make difficult problems in trigonometry, calculus, and physics much easier to solve. They are essential tools for rewriting and manipulating equations into a more manageable form.
7. Can you give an example of how trigonometric identities are used in real life?
Yes. In physics and engineering, trigonometric identities are crucial for analysing anything that behaves like a wave. This includes sound waves, light waves, and AC electrical circuits. For example, when multiple waves interact, their combined effect is described by an expression that is often simplified using identities to predict the resulting wave's amplitude and phase.
8. How do reciprocal identities (like secθ = 1/cosθ) relate to the main Pythagorean identities?
Reciprocal identities act as a bridge, allowing you to use Pythagorean identities in more situations. For example, the identity 1 + tan²θ = sec²θ is derived by taking the fundamental identity, sin²θ + cos²θ = 1, and dividing every term by cos²θ. This process directly uses the definitions tanθ = sinθ/cosθ and secθ = 1/cosθ. They provide flexible pathways to connect different trigonometric ratios within a proof.
9. What is the difference between proving an identity and verifying it?
There's a crucial mathematical distinction:
Proving an identity is a formal, logical argument. It involves using algebra and established identities to show that the equation holds true for all possible angles.
Verifying an identity means substituting one or more specific angles (e.g., 30°, 45°) to check if the equation works for those values. While verification can help spot errors, it is not a proof because it doesn't guarantee the identity works for every other angle.
10. Are the Class 10 trigonometric identities only valid for acute angles?
While in Class 10, these identities are introduced using acute angles (0° to 90°) within a right-angled triangle, they are in fact universally true. The relationships they describe are fundamental properties of the trigonometric functions themselves, which means they apply to any angle, including angles greater than 90°, negative angles, and angles measured in radians.
