Class 9 Maths Exercise 6.4: Trigonometry & Bearings

Class 9 Mathematics Exercise 6.4 – Trigonometry Identities and Equations (Solved with Explanation)

Exercise 6.4 of Class 9 Mathematics focuses on one of the most important parts of trigonometry: proving identities and solving trigonometric equations. This exercise builds the foundation for higher-level mathematics and frequently appears in board exams.

In this lesson, you will learn how to simplify expressions, prove identities step by step, and solve equations involving sine, cosine, tangent, and their related functions.

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Core Concepts Used in Exercise 6.4

Before solving questions, you must understand the following key identities:

• Pythagorean Identity: sin²θ + cos²θ = 1
• Reciprocal Identities: secθ = 1/cosθ, cscθ = 1/sinθ
• Quotient Identities: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
• Algebraic Identity: (a + b)(a − b) = a² − b²

From experience, most students lose marks not because they do not know the formulas, but because they do not know when and how to apply them. This exercise trains you to think step by step.

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How to Prove Trigonometric Identities

When proving identities, always start with one side (usually the left-hand side) and simplify it until it becomes equal to the right-hand side.

Example 1

Prove that:

1 − sin²θ · sec²θ = 1

Solution:

• We know: 1 − sin²θ = cos²θ
• Also: sec²θ = 1 / cos²θ

Substitute:

cos²θ × (1 / cos²θ) = 1

Hence, proved.

Key Insight: Always try to convert everything into sine and cosine first. This makes simplification much easier.

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Example 2

Prove that:

(1 − sinθ) / (1 + sinθ) = (secθ − tanθ)²

Solution Strategy:

• Multiply numerator and denominator by (1 − sinθ)
• Use identity: 1 − sin²θ = cos²θ
• Convert into sec and tan form

After simplification, both sides become equal.

Exam Tip: Whenever you see fractions, try multiplying by conjugates to simplify.

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Handling Square Roots in Trigonometry

Square root expressions often confuse students. The key idea is to remove the square root by multiplying appropriately.

Example 3

Prove that:

√((1 − cosθ) / (1 + cosθ)) = sinθ / (1 + cosθ)

Method:

• Multiply numerator and denominator by (1 + cosθ)
• Use identity: (1 − cos²θ) = sin²θ
• Take square root carefully

This simplifies the expression and proves the identity.

Common Mistake: Students often forget that √(sin²θ) = sinθ (not ±sinθ in this level).

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Advanced Identities (tan, cot, sec, csc)

Many questions in Exercise 6.4 combine multiple identities. You must be comfortable switching between forms.

Example 4

Show that:

sec²θ + csc²θ = (tanθ + cotθ)²

Approach:

• Convert sec²θ = 1/cos²θ and csc²θ = 1/sin²θ
• Take common denominator
• Simplify carefully

After simplification, both sides match.

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Algebraic Trigonometry (Important for Exams)

Some questions mix algebra and trigonometry. These are high-scoring if done correctly.

Example 5

Given:

x cosθ + y sinθ = m x sinθ − y cosθ = n

Prove that:

x² + y² = m² + n²

Solution:

• Square both equations
• Add them together
• Middle terms cancel out

Result:

x² + y² = m² + n²

Insight: This method (squaring and adding) is very important and appears frequently in exams.

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Solving Trigonometric Equations

The final part of this exercise involves finding angles.

Example 6

Solve:

2 sin²θ = 1/2

Solution:

• sin²θ = 1/4
• sinθ = 1/2
• θ = 30°

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Example 7

Solve:

sinθ = cosθ

Solution:

• Divide both sides by cosθ
• tanθ = 1
• θ = 45°

Exam Tip: Always reduce equations to basic ratios like tanθ, sinθ, or cosθ.

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Why Exercise 6.4 Is Important

This exercise is important because it develops your ability to:

• Simplify complex expressions
• Prove identities logically
• Apply formulas correctly
• Solve equations step by step

From teaching experience, students who master this exercise find Chapter 6 much easier and score higher in board exams.

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Common Mistakes to Avoid

• Skipping steps while proving identities
• Using wrong identities
• Not converting everything into sine and cosine
• Ignoring algebraic simplification
• Making sign errors in equations

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Final Preparation Tips

• Practice each identity multiple times
• Focus on step-by-step presentation
• Memorize core identities properly
• Always verify your final answer

Strong concepts and clear working will help you secure full marks in trigonometry questions.