All Trigonometric Identities Explained Simply (Class 10 Guide)
Trigonometry is one of the most important chapters in Class 10 Mathematics. Understanding trigonometric identities helps students simplify complex expressions and solve mathematical problems quickly during exams.
Many students struggle with remembering formulas or understanding when to apply them. This guide explains all trigonometric identities in a simple and logical way with step-by-step solved problems so that you can master the topic easily.
What are Trigonometric Identities?
A trigonometric identity is an equation that is true for all values of the variable where both sides of the equation are defined.
For example:
sin²θ + cos²θ = 1
This identity is always true regardless of the value of θ. Trigonometric identities are extremely useful because they allow us to transform complicated expressions into simpler forms.
Basic Trigonometric Ratios
Before learning identities, we must understand the six basic trigonometric ratios.
- sin θ = Perpendicular / Hypotenuse
- cos θ = Base / Hypotenuse
- tan θ = Perpendicular / Base
- cosec θ = Hypotenuse / Perpendicular
- sec θ = Hypotenuse / Base
- cot θ = Base / Perpendicular
These ratios are the foundation of all trigonometric identities.
1. Reciprocal Identities
Reciprocal identities show the relationship between a trigonometric function and its reciprocal.
- sin θ = 1 / cosec θ
- cos θ = 1 / sec θ
- tan θ = 1 / cot θ
- cosec θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
Example 1: If sin θ = 1/2, find cosec θ.
cosec θ = 1 / sin θ
= 1 / (1/2)
= 2
Answer: cosec θ = 2
2. Quotient Identities
Quotient identities express tangent and cotangent in terms of sine and cosine.
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Example 2: If sin θ = 3/5 and cos θ = 4/5, find tan θ.
tan θ = sin θ / cos θ
= (3/5) ÷ (4/5)
= 3/4
Answer: tan θ = 3/4
3. Pythagorean Identities
These are the most important identities in trigonometry and are derived from the Pythagorean theorem.
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Example 3: If sin θ = 3/5, find cos θ.
Using identity:
sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 16/25
cos θ = 4/5
4. Trigonometric Identities Used for Simplification
These identities help simplify algebraic expressions containing trigonometric functions.
Identity 1
(1 − sin²θ) = cos²θ
Identity 2
(1 − cos²θ) = sin²θ
Identity 3
(sec²θ − 1) = tan²θ
Identity 4
(cosec²θ − 1) = cot²θ
Example 4: Simplify (1 − sin²θ) / cos²θ
Using identity:
1 − sin²θ = cos²θ
Expression becomes:
cos²θ / cos²θ = 1
Answer = 1
5. Standard Trigonometric Values
Students should memorize the following values because they are used frequently in exams.
- sin 0° = 0
- sin 30° = 1/2
- sin 45° = 1/√2
- sin 60° = √3/2
- sin 90° = 1
- cos 0° = 1
- cos 30° = √3/2
- cos 45° = 1/√2
- cos 60° = 1/2
- cos 90° = 0
Solved Trigonometric Identity Problems
Problem 1
Simplify:
(sin θ / cos θ) × (cos θ / sin θ)
= tan θ × cot θ
= 1
Problem 2
Prove that:
(1 + tan²θ) / sec²θ = 1
Using identity:
1 + tan²θ = sec²θ
Therefore:
sec²θ / sec²θ = 1
Problem 3
Simplify:
sin²θ / (1 − cos²θ)
Using identity:
1 − cos²θ = sin²θ
Expression becomes:
sin²θ / sin²θ = 1
Problem 4
If tan θ = 5/12, find sin θ and cos θ.
Consider a right triangle:
Perpendicular = 5
Base = 12
Hypotenuse = √(5² + 12²)
= √169 = 13
sin θ = 5/13
cos θ = 12/13
Exam Tips for Trigonometric Identities
✔ Memorize Pythagorean identities because they appear frequently in board exams.
✔ Cancel common factors carefully to avoid mistakes.
✔ Practice at least 10 identity problems daily before exams.
Common Mistakes Students Make
- Mixing reciprocal identities
- Forgetting to square trigonometric functions
- Using wrong Pythagorean identity
- Incorrect simplification of fractions
Why Trigonometric Identities are Important
Trigonometric identities are not only important for Class 10 board exams but also for higher mathematics, physics, engineering, and competitive exams. They simplify complex trigonometric expressions and help in solving equations efficiently.
