CBSE Class 10th Trigonometric Identities Details & Preparations Downloads

In the vast landscape of mathematics, trigonometry stands as a captivating branch that unveils the intricate relationships between angles and sides in triangles. At the heart of this fascinating domain lie trigonometric identities – mathematical expressions that link various trigonometric functions in harmonious ways. Join us on a journey into the world of trigonometric identities, where precision meets elegance, and equations dance to the rhythm of mathematical beauty.

Unlocking Mathematical Brilliance CBSE NCERT Trigonometric Identities – Your Essential Download Guide

What are Trigonometric Identities?

Trigonometric Identities are the equalities that involve trigonometry functions and hold true for all the values of variables given in the equation. There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.

All the trigonometric identities are based on the six trigonometric ratios. They are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from the six trigonometric ratios.

List of Trigonometric Identities

There are various identities in trigonometry that are used to solve many trigonometric problems. Using these trigonometric identities or formulas, complex trigonometric questions can be solved quickly. Let us see all the fundamental trigonometric identities here.
Reciprocal Trigonometric Identities

The reciprocal trigonometric identities are

• Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
• Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
• Tan θ = 1/Cot θ or Cot θ = 1/Tan θ

Pythagorean Trigonometric Identities

There are three Pythagorean trigonometric identities in trigonometry that are based on the right-triangle theorem or Pythagoras theorem.

• sin² a + cos² a = 1
• 1+tan² a  = sec² a
• cosec² a = 1 + cot² a

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The Foundation: Basic Trigonometric Functions

Before we delve into the realm of identities, it's essential to grasp the fundamental trigonometric functions: sine, cosine, and tangent. These functions, originating from the ratios of sides in a right-angled triangle, lay the groundwork for the identities The Pythagorean identity is a cornerstone in the world of trigonometry. Derived from the Pythagorean theorem, it expresses a profound relationship between the sine and cosine functions: This simple yet powerful identity forms the basis for many other trigonometric identities and plays a crucial role in trigonometric manipulations.

 Reciprocal and Quotient Identities: Interweaving Functions

Trigonometric identities extend beyond the Pythagorean realm to include reciprocal and quotient identities. These expressions reveal the interconnected nature of trigonometric functions: Understanding these identities opens up avenues for simplifying complex trigonometric expressions and solving trigonometric equations with ease.

Double Angle and Half Angle Identities: The Symmetry of Mathematics

Double-angle and half-angle identities bridge the gap between different angles, offering a symphony of relations that simplify trigonometric expressions: These identities not only showcase the elegance of mathematical relationships but also serve as valuable tools in problem-solving and calculus.

Sum and Difference Identities: A Mathematical Ballet

Sum and difference identities provide a captivating dance of trigonometric functions, expressing the trigonometric values of the sum or difference of two angles:

Ratio Trigonometric Identities

The trigonometric ratio identities are:

• Tan θ = Sin θ/Cos θ
• Cot θ = Cos θ/Sin θ

Trigonometric Identities of Opposite Angles

The list of opposite angle trigonometric identities are:

• Sin (-θ) = – Sin θ
• Cos (-θ) = Cos θ
• Tan (-θ) = – Tan θ
• Cot (-θ) = – Cot θ
• Sec (-θ) = Sec θ
• Csc (-θ) = -Csc θ

Trigonometric Identities of Complementary Angles

In geometry, two angles are complementary if their sum is equal to 90 degrees. Similarly, we can learn here the trigonometric identities for complementary angles.

• Sin (90 – θ) = Cos θ
• Cos (90 – θ) = Sin θ
• Tan (90 – θ) = Cot θ
• Cot ( 90 – θ) = Tan θ
• Sec (90 – θ) = Csc θ
• Csc (90 – θ) = Sec θ

Trigonometric Identities of Supplementary Angles

Two angles are supplementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for supplementary angles.

• sin (180°- θ) = sinθ
• cos (180°- θ) = -cos θ
• cosec (180°- θ) = cosec θ
• sec (180°- θ)= -sec θ
• tan (180°- θ) = -tan θ
• cot (180°- θ) = -cot θ

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SAMPLE PRACTICE QUESTION

Q1: Prove the trigonometric identity: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)?

Ans: To prove \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), we use the sum-to-product formula for sine.

Q2: Simplify the expression: \( \frac{\tan\theta}{1-\cot\theta} + \frac{\cot\theta}{1-\tan\theta} \)?

Ans: Combine fractions and simplify using trigonometric identities to get the final expression in terms of sine and cosine.

Q3: Prove the sum-to-product identity: \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)?

Ans: Apply sum-to-product formulas for sine and simplify step by step to establish the given identity

Q4: Evaluate \( \tan\frac{\pi}{3} \cdot \tan\frac{\pi}{4} \cdot \tan\frac{\pi}{6} \)?

Ans: Use trigonometric identities and angle sum/difference formulas to simplify and find the value of the given expression.

Q5: Derive the triple angle formula for cosine: \( \cos(3\theta) = 4\cos^3\theta - 3\cos\theta \)?

Ans: Apply the double angle formula multiple times, and simplify to express the cosine of triple angle in terms of the cosine of the original angle.

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