Trigonometric Identities | The Concepts Adda

01 · THE BASICS

What is a Trigonometric Identity?

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it holds true for all values of the angle θ for which the ratios are defined.

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Key Idea

Trigonometric identities are derived directly from the Pythagoras Theorem applied to a right-angled triangle.

02 · THREE FUNDAMENTAL IDENTITIES

The 3 Key Identities

IDENTITY 1

sin² θ + cos² θ = 1

Useful rearrangements:

sin² θ = 1 - cos² θ
cos² θ = 1 - sin² θ

IDENTITY 2

1 + tan² θ = sec² θ

Useful rearrangements:

sec² θ - tan² θ = 1
tan² θ = sec² θ - 1

IDENTITY 3

1 + cot² θ = cosec² θ

Useful rearrangements:

cosec² θ - cot² θ = 1
cot² θ = cosec² θ - 1

03 · STEP-BY-STEP PROOFS

Geometric Proofs of the Identities

All three Pythagorean identities are derived from a right-angled triangle ΔABC (right-angled at B) with angle θ = ∠C by applying the Pythagoras Theorem and dividing by each side.

Reference Right-Angled Triangle

In ΔABC, ∠B = 90° and ∠C = θ:

Perpendicular (p) = AB (side opposite to θ)
Base (b) = BC (side adjacent to θ)
Hypotenuse (h) = AC (side opposite to 90°)

By Pythagoras Theorem: AB² + BC² = AC² — (Equation 1)

PROOF 1

Proof of sin² θ + cos² θ = 1

Step 1

Start with the Pythagoras Theorem equation for ΔABC:

AB² + BC² = AC²

Step 2

Divide both sides of Equation 1 by AC² (Hypotenuse squared):

(AB² / AC²) + (BC² / AC²) = AC² / AC²

(AB / AC)² + (BC / AC)² = 1

Step 3

Substitute trigonometric ratio definitions:

Since sin θ = AB / AC and cos θ = BC / AC:

Substituting these ratios into the equation yields:

(sin θ)² + (cos θ)² = 1 ⟹ sin² θ + cos² θ = 1 (Proved)

PROOF 2

Proof of 1 + tan² θ = sec² θ

Step 1

Start with the Pythagoras Theorem equation:

AB² + BC² = AC²

Step 2

Divide both sides of Equation 1 by BC² (Base squared):

(AB² / BC²) + (BC² / BC²) = AC² / BC²

(AB / BC)² + 1 = (AC / BC)²

Step 3

Substitute trigonometric ratio definitions:

Since tan θ = AB / BC and sec θ = AC / BC:

Substituting these ratios yields:

(tan θ)² + 1 = (sec θ)² ⟹ 1 + tan² θ = sec² θ (Proved)

Note: Valid for 0° ≤ θ < 90° (where cos θ ≠ 0)

PROOF 3

Proof of 1 + cot² θ = cosec² θ

Step 1

Start with the Pythagoras Theorem equation:

AB² + BC² = AC²

Step 2

Divide both sides of Equation 1 by AB² (Perpendicular squared):

(AB² / AB²) + (BC² / AB²) = AC² / AB²

1 + (BC / AB)² = (AC / AB)²

Step 3

Substitute trigonometric ratio definitions:

Since cot θ = BC / AB and cosec θ = AC / AB:

Substituting these ratios yields:

1 + (cot θ)² = (cosec θ)² ⟹ 1 + cot² θ = cosec² θ (Proved)

Note: Valid for 0° < θ ≤ 90° (where sin θ ≠ 0)

04 · PRACTICE PROBLEMS

Test Your Understanding

EXAMPLE 1

Prove: (1 - sin² θ) sec² θ = 1

Solution:

Take LHS = (1 - sin² θ) sec² θ

By Identity 1: 1 - sin² θ = cos² θ

Also, sec² θ = 1 / cos² θ

LHS = cos² θ × (1 / cos² θ)
= 1 = RHS (Proved)

EXAMPLE 2

Prove: (sec A + tan A)(1 - sin A) = cos A

Solution:

Convert everything to sin and cos:

sec A = 1/cos A and tan A = sin A/cos A

LHS = ( (1 + sin A) / cos A ) × (1 - sin A)

= (1 - sin² A) / cos A

Since 1 - sin² A = cos² A:
LHS = cos² A / cos A = cos A = RHS (Proved)

04 · QUICK SUMMARY

Quick Revision

sin² θ + cos² θ = 1 is the primary Pythagorean identity.

sec² θ - tan² θ = 1 for all 0° ≤ θ < 90°.

cosec² θ - cot² θ = 1 for all 0° < θ ≤ 90°.