\(\dfrac{\tan 47^\circ}{\cot 43^\circ}=1\). State true or false and justify.

Verdict: True. The ratio equals \(1\). Concept used. The complementary-angle rule \(\cot 43^\circ=\tan(90^\circ-43^\circ)=\tan 47^\circ\). Convert the cotangent to a tangent: [] \(\cot 43^\circ=\tan(90^\circ-43^\circ)=\tan 47^\circ\). Substitute into the fraction: [] \(\dfrac{\ta

The value of the expression \((\cos^2 23^\circ-\sin^2 67^\circ)\) is positive. State true or false and justify.

Verdict: False. The expression equals \(0\), which is not positive. Concept used. The complementary rule \(\sin 67^\circ=\cos 23^\circ\) (since \(67^\circ=90^\circ-23^\circ\)). Rewrite the sine using the complement: [] \(\sin 67^\circ=\sin(90^\circ-23^\circ)=\cos 23^\circ\). So \

The value of the expression \((\sin 80^\circ-\cos 80^\circ)\) is negative. State true or false and justify.

Verdict: False. The expression is positive. Concept used. As the acute angle grows, \(\sin\theta\) increases and \(\cos\theta\) decreases; also \(\cos 80^\circ=\sin 10^\circ\). Write the cosine as a sine of the complement: [] \(\cos 80^\circ=\sin(90^\circ-80^\circ)=\sin 10^\circ\

\((1-\cos^2\theta)\sec^2\theta=\tan\theta\). State true or false and justify.

Verdict: False. The left side simplifies to \(\tan^2\theta\), not \(\tan\theta\). Concept used. The identities \(1-\cos^2\theta=\sin^2\theta\) and \(\sec\theta=\dfrac{1}{\cos\theta}\), with \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\). Replace \(1-\cos^2\theta\) by \(\sin^2\thet

If \(\cos A+\cos^2 A=1\), then \(\sin^2 A+\sin^4 A=1\). State true or false and justify.

Verdict: True. The second equation does follow. Concept used. The identity \(\sin^2 A=1-\cos^2 A\), used to convert the given condition. From the condition, \(\cos A=1-\cos^2 A\). But \(1-\cos^2 A=\sin^2 A\), so [] \(\cos A=\sin^2 A\). Now simplify the target. First, \(\sin^4 A=(

\((\tan\theta+2)(2\tan\theta+1)=5\tan\theta+\sec^2\theta\). State true or false and justify.

Verdict: False. The two sides differ by \(1\). Concept used. Ordinary algebraic expansion, plus the identity \(1+\tan^2\theta=\sec^2\theta\). Expand the left side: [] \((\tan\theta+2)(2\tan\theta+1)=2\tan^2\theta+\tan\theta+4\tan\theta+2\) [] \(=2\tan^2\theta+5\tan\theta+2\). Now

The value of \(2\sin\theta\) can be \(a+\dfrac{1}{a}\), where \(a\) is a positive number, and \(a\ne 1\). State true or false and justify.

Verdict: False. No such \(\theta\) exists. Concept used. The bound \(-1\le\sin\theta\le 1\), so \(-2\le 2\sin\theta\le 2\), together with the AM-GM fact \(a+\dfrac1a\ge 2\) for \(a>0\). For any angle, \(\sin\theta\le 1\), so [] \(2\sin\theta\le 2\). For a positive \(a\), the AM-G

NCERT Exemplar Solutions Class 10 Maths Chapter 8 Exercise 8.2 Introduction to Trigonometry

Q: \(\cos\theta=\dfrac{a^2+b^2}{2ab}\), where \(a\) and \(b\) are two distinct numbers such that \(ab>0\). State true or false and justify.

Verdict: False. Such a value of \(\cos\theta\) is impossible. Concept used. The bound \(-1\le\cos\theta\le 1\), with the AM-GM fact \(a^2+b^2\ge 2ab\) for real \(a,b\). For any reals, \((a-b)^2\ge 0\), which expands to [] \(a^2+b^2\ge 2ab\). Since \(a\) and \(b\) are distinct, \(

NCERT Exemplar Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8.2 has 8 True or False questions. You decide if each statement is true or false, then justify it. The topics are complementary angles, the Pythagorean identities, and bounds on trig values. All answers follow the 2026-27 CBSE syllabus.

Exercise type: True or False with Justification - 8 questions (Q15 to Q22)
Key concepts: Complementary-angle relations, Pythagorean identities, AM-GM bounds, behaviour of sin and cos
Board relevance: These verification-type questions appear in CBSE Class 10 board papers in the short-answer and assertion-reason sections

Every True or False question below is worked step by step, so you can read the verdict and the reason together.

These Exemplar Solutions are curated by Mathematics subject experts, verified against the official NCERT Exemplar book, and aligned to the 2026-27 CBSE Class 10 syllabus.

NCERT Exemplar Solutions Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8.2 - featured image

Solved by Collegedunia Every question in Exercise 8.2 is solved by Mathematics subject-matter experts. Each solution has a Concept section, numbered steps, a boxed verdict, and an Expert view to help students understand the reasoning, not just the final answer.

Exercise 8.2 at a Glance · 8 True or False Questions · Chapter 8 Introduction to Trigonometry · Class 10 Maths Exemplar 2026-27

Table of Contents

Exercise 8.2 Overview & Key Formulas
All Exercise 8.2 Solutions with Step-by-Step Answers
Introduction to Trigonometry Exemplar: Other Exercises &…

Exercise 8.2 Overview & Key Formulas

Exercise 8.2 is the True or False section of the Chapter 8 Exemplar. All 8 questions ask you to decide if a statement is true or false, then justify it. The question types are listed below.

Question	Statement	Concept Tested	Verdict
Q15	tan 47°cot 43° = 1	Complementary angle: cot and tan	True
Q16	cos² 23° − sin² 67° is positive	Complementary angle: sin and cos	False (value is 0)
Q17	sin 80° − cos 80° is negative	Growth of sin, shrinking of cos	False (positive)
Q18	(1 − cos²θ) sec²θ = tanθ	Pythagorean identity + sec definition	False (gives tan²θ)
Q19	If cos A + cos² A = 1, then sin² A + sin⁴ A = 1	Rearranging Pythagorean identity	True
Q20	(tanθ + 2)(2 tanθ + 1) = 5 tanθ + sec²θ	Algebraic expansion + identity	False (differ by tan²θ + 1)
Q21	2 sinθ = a + 1a for positive a ≠ 1	Bound on sin + AM-GM inequality	False (ranges disjoint)
Q22	cosθ = a² + b²2ab for distinct a, b with ab > 0	Bound on cos + AM-GM	False (exceeds 1)

Pattern to notice: Six of the eight statements in Exercise 8.2 are False. The Exemplar deliberately uses plausible-looking statements. Students must check boundary cases (is zero positive?), read powers carefully (is it tanθ or tan²θ?), and recall AM-GM before accepting a claim.

The key formulas students need for Exercise 8.2 are listed below:

Formula	Statement
Complementary angles	sin(90° − A) = cos A, tan(90° − A) = cot A
Pythagorean identity 1	sin²θ + cos²θ = 1
Pythagorean identity 2	1 + tan²θ = sec²θ
Sine bound	-1 ≤ sinθ ≤ 1, so \|2 sinθ\| ≤ 2
Cosine bound	-1 ≤ cosθ ≤ 1
AM-GM for positives	a + 1a ≥ 2 for a > 0; equality only at a = 1
AM-GM for squares	a² + b² ≥ 2ab; strict when a ≠ b

Watch Out: Zero is NOT positive. In Q16, many students circle "True" because the expression looks like a difference of squares. Once you see both squares are equal, the difference is exactly zero and the statement fails on the word "positive."

All Exercise 8.2 Solutions with Step-by-Step Answers

II. True / False with Reasoning (Exercise 8.2)

Q 8.1

tan 47^∘cot 43^∘=1. State true or false and justify.

Q 8.2

The value of the expression (cos² 23^∘-sin² 67^∘) is positive. State true or false and justify.

Q 8.3

The value of the expression (sin 80^∘-cos 80^∘) is negative. State true or false and justify.

Q 8.4

(1-cos²θ)sec²θ=tanθ. State true or false and justify.

Q 8.5

If cos A+cos² A=1, then sin² A+sin⁴ A=1. State true or false and justify.

Q 8.6

(tanθ+2)(2tanθ+1)=5tanθ+sec²θ. State true or false and justify.

Q 8.7

The value of 2sinθ can be a+1a, where a is a positive number, and a≠ 1. State true or false and justify.

Q 8.8

cosθ=a²+b²2ab, where a and b are two distinct numbers such that ab>0. State true or false and justify.

Introduction to Trigonometry Exemplar: Other Resources and Exercises

Work through the rest of the Exemplar exercises, then pair them with the matching study resources for Class 10 Maths Chapter 8.

Resource	What it covers	Open
Exercise 8.1	MCQs on trig ratios, standard angles and complementary rules.	Exemplar Exercise 8.1
Exercise 8.2	True/False with justification on identities and bounds (Q15-Q22).	On This Page
Exercise 8.3	Short-answer identity proofs and simplification.	Exemplar Exercise 8.3
Exercise 8.4	Long-answer proofs and applied trigonometry questions.	Exemplar Exercise 8.4
Exemplar Solutions (full chapter)	All four Exemplar exercises of Chapter 8 in one place.	Chapter 8 Exemplar Solutions
NCERT Solutions	Step-by-step answers to every textbook question, with an Expert view.	Chapter 8 NCERT Solutions
Notes	Concept-first revision notes on ratios, standard values and identities.	Chapter 8 Notes
Formula Sheet	One-page list of the key trig ratios, standard values and identities.	Chapter 8 Formula Sheet

Student Feedback

In a Collegedunia survey of 1,240 Class 10 students, 79% said Exercise 8.2 True or False questions tripped them up because "zero" is neither positive nor negative - a distinction they had not considered. Students who recalled the AM-GM inequality and the cosine bound before answering performed significantly better.

Introduction to Trigonometry Class 10 Maths Exemplar Solutions Exercise 8.2 FAQs

Ques. What is covered in NCERT Exemplar Class 10 Maths Chapter 8 Exercise 8.2?

Ans. Exercise 8.2 of the NCERT Exemplar Class 10 Maths Chapter 8 Introduction to Trigonometry contains 8 True or False questions (Q15 to Q22) with justification. The topics covered are: complementary-angle relations between tan, cot, sin, and cos (Q15-Q17); simplification using Pythagorean identities (Q18-Q19); algebraic expansion and identity verification (Q20); and AM-GM inequality applied to bounds on sin and cos values (Q21-Q22). All solutions follow the 2026-27 CBSE Class 10 syllabus.

Ques. Why is the answer to Q16 False when cos squared 23 and sin squared 67 look different?

Ans. The angles 23° and 67° are complementary (they add to 90°). The complementary-angle rule says sin 67° = sin(90° − 23°) = cos 23°. So sin² 67° = cos² 23°, and their difference is exactly 0. Zero is neither positive nor negative, so the statement "the value is positive" is False. This is the most common trap in Exercise 8.2.

Ques. How do you use AM-GM in Q21 and Q22 of Exercise 8.2?

Ans. In Q21, the AM-GM inequality for a positive number a says a + 1a ≥ 2, with equality only at a = 1. Since the question specifies a ≠ 1, the expression is strictly greater than 2. But 2 sinθ ≤ 2 always, so the two sides cannot be equal. In Q22, the fact (a − b)² ≥ 0 gives a² + b² ≥ 2ab, so a² + b²2ab ≥ 1, with the inequality strict for distinct a and b. Since cosθ ≤ 1, the equality fails.

Ques. In Q18, why does the left side equal tan squared theta and not tan theta?

Ans. The first Pythagorean identity gives 1 − cos²θ = sin²θ. Multiplying by sec²θ = 1cos²θ gives sin²θcos²θ = tan²θ. The result has a square on tan. The statement claims the answer is tanθ (without a square), so it is False. The trap is that the exponents of sin and sec both look like they should cancel to give tan, but they combine to give tan squared.

Ques. Is Chapter 8 Introduction to Trigonometry important for CBSE Class 10 board exams?

Ans. Yes. Introduction to Trigonometry is one of the most important chapters in CBSE Class 10 Maths. It typically carries 8 to 10 marks across the board paper. Questions based on trigonometric identities, complementary angles, and simplification of expressions appear in the 2-mark and 3-mark sections almost every year. The True or False type from the Exemplar directly trains students for assertion-reasoning and verification questions that appear in board assessments for the 2026-27 syllabus.