Class 10 Maths Chapter 7 Exercise 7.3 Solutions 2026 – Trigonometry Notes PDF

Unit 7: Trigonometry | Exercise 7.3 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 7.3 solutions for Trigonometry have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Equations True for Every Angle

A trigonometric identity is different from an ordinary equation: instead of being true for just one or two specific values of θ, it holds for every possible angle. Exercise 7.3 introduces the three fundamental Pythagorean identities, derived directly from applying the Pythagorean theorem to the six ratios from Exercise 7.2.

Key Concepts Covered in Exercise 7.3

The First Pythagorean Identity

sin²θ + cos²θ = 1 is the most fundamental trigonometric identity, derived directly from the Pythagorean theorem applied to a right triangle with hypotenuse 1.

The Second Pythagorean Identity

Dividing the first identity by cos²θ gives 1 + tan²θ = sec²θ, relating the tangent and secant ratios.

The Third Pythagorean Identity

Dividing the first identity by sin²θ instead gives 1 + cot²θ = cosec²θ, the parallel identity for cotangent and cosecant.

Proving Identities

This exercise gives practice using the three Pythagorean identities (often combined with basic algebra) to prove that one trigonometric expression equals another, usually by simplifying one side until it matches the other.

Step-by-Step Solved Examples

Q. No. 1: Simplify sin²θ + cos²θ + 5.

Using sin²θ + cos²θ = 1:

= 1 + 5 = 6

Q. No. 2: Prove that sec²θ − tan²θ = 1.

Start from 1 + tan²θ = sec²θ

Rearranging: sec²θ − tan²θ = 1 ✔ proved directly by rearrangement

Q. No. 3: Simplify (1 − sin²θ) sec²θ.

Using 1 − sin²θ = cos²θ (rearranged first identity):

= cos²θ × sec²θ

Since secθ = 1/cosθ, sec²θ = 1/cos²θ:

= cos²θ × (1/cos²θ) = 1

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 7.3 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions on stating one of the three Pythagorean identities
• Short Questions: Simplifying a short expression using an identity, like Q. No. 1 and 3 above
• Long Questions: Proving a more involved identity through several algebraic steps

Where This Matters Beyond This Exercise

These identities are the algebraic toolkit used throughout the rest of the chapter and in Intermediate-level calculus, where simplifying trigonometric expressions is a routine step in solving equations and integrals. Recognizing when an identity applies is often the key that unlocks an otherwise difficult-looking problem.

Common Mistakes Students Make in Exercise 7.3

• Writing sin²θ as sinθ² (squaring the angle instead of the ratio)
• Forgetting which identity pairs with which ratio (mixing up tan/sec with cot/cosec)
• Trying to prove an identity by assuming it’s true and working from both sides simultaneously, which isn’t valid proof technique
• Making an algebraic slip while rearranging an identity into a different form

Why This Exercise Matters for the Board Exam

Proving trigonometric identities is one of the most heavily weighted long-question types in this chapter, since it tests both memorized formulas and algebraic manipulation skill together. Presenting each step clearly, stating which identity is used at each stage, tends to earn strong method marks even if the final simplification takes a few attempts.

Quick Links – Chapter 7: Trigonometry

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Download Exercise 7.3 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 7.3 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. How many trigonometric identities do I need to memorize here?

This exercise focuses on the three Pythagorean identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, and 1+cot²θ=cosec²θ. All three come from the same underlying relationship, so learning the first well makes the other two easier to derive or recall.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

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