Evaluate the following: (i) \(\sin60^\circ\cos30^\circ+\sin30^\circ\cos60^\circ\) (ii) \(2\tan^2 45^\circ+\cos^2 30^\circ-\sin^2 60^\circ\) (iii) \(\dfrac{\cos45^\ci

Concept used. Every term is a specific-angle value that we substitute from the standard table: \(\sin30^\circ=\tfrac12,\ \cos30^\circ=\tfrac{\sqrt3}{2},\ \sin60^\circ=\tfrac{\sqrt3}{2},\ \cos60^\circ=\tfrac12,\ \tan45^\circ=1,\ \cot45^\circ=1,\ \sec30^\circ=\tfrac{2}{\sqrt3},\ \c

Choose the correct option and justify your choice: (i) \(\dfrac{2\tan30^\circ}{1+\tan^2 30^\circ}=\) (A) \(\sin60^\circ\) (B) \(\cos60^\circ\) (C) \

Concept used. Each part is evaluated by substituting the specific-angle values \(\tan30^\circ=\tfrac{1}{\sqrt3}\) and \(\tan45^\circ=1\), then matching the result to a known value. (i) With \(\tan30^\circ=\tfrac{1}{\sqrt3}\), so \(\tan^2 30^\circ=\tfrac13\): \[ \frac{2\cdot\tfrac

If \(\tan(A+B)=\sqrt3\) and \(\tan(A-B)=\dfrac{1}{\sqrt3}\); \(0^\circ B\), find \(A\) and \(B\).

Concept used. We match each given tangent to a specific-angle value to turn the trigonometric equations into ordinary linear equations in \(A\) and \(B\), then solve the pair. \(\tan(A+B)=\sqrt3=\tan60^\circ\), and since \(0^\circ \(\tan(A-B)=\tfrac{1}{\sqrt3}=\tan30^\circ\), and

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

Q: State whether the following are true or false. Justify your answer. (i) \(\sin(A+B)=\sin A+\sin B\). (ii) The value of \(\sin\theta\) increases as \(\theta\) increases. (

Concept used. We test each statement using the standard table values and the behaviour of the ratios as \(\theta\) runs from \(0^\circ\) to \(90^\circ\): \(\sin\theta\) rises from \(0\) to \(1\), \(\cos\theta\) falls from \(1\) to \(0\), and \(\cot0^\circ=\tfrac{1}{\tan0^\circ}\)

The NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8.2 cover all 4 questions on evaluating trigonometric expressions at specific angles, multiple-choice reasoning, finding unknown angles, and true/false justifications using the standard value table. Every answer is solved step by step for the 2026-27 CBSE syllabus.

Questions covered: 4 in total (Q1 to Q4), ranging from evaluating expressions using sin 30°, cos 45°, tan 60° values to reasoning about how sin and cos behave as the angle increases.
Core method: substitute standard table values, simplify arithmetic, and rationalise any surds left in the denominator.
CBSE board value: Exercise 8.2 questions carry 2 to 3 marks and test the Pythagorean identity sin²A + cos²A = 1 and the double-angle pattern alongside direct substitution.

Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8.2 NCERT Solutions - specific angle values and identities

Solved by Collegedunia: Every Exercise 8.2 question below is solved by Mathematics subject experts, checked against the official 2026-27 NCERT textbook, and written with full working so each substitution step, simplification, and rationalisation earns its marks in the CBSE Class 10 board paper.

What Exercise 8.2 of Introduction to Trigonometry Covers for Class 10

Exercise 8.2 is the specific-angle values exercise of Chapter 8. After learning trigonometric ratios from the side lengths of a right triangle in Exercise 8.1, students now use the standard table of exact values for 0°, 30°, 45°, 60°, and 90° to evaluate and reason about expressions.

Q1 (5 parts): evaluate expressions involving sin 60°, cos 30°, tan 45°, sec 30°, cosec 30°, cosec 60°, cos 45°, and the Pythagorean identity. Parts (iii) and (iv) require rationalising surds.
Q2 (4 parts MCQ): choose the correct option and justify using the double-angle forms (recognising sin 2θ, cos 2θ, tan 2θ patterns without needing the formula).
Q3: solve for two unknowns A and B from a pair of tangent equations using the standard table.
Q4 (5 parts): state true or false and justify each claim about the behaviour of sine, cosine, cotangent, and the additivity of sine.

Standard Trigonometric Values Table Used Throughout Exercise 8.2

Every question in Exercise 8.2 reduces to substituting values from this table. Memorising it is the single most important step before attempting any part of this exercise.

Angle	sin	cos	tan	cosec	sec	cot
0°	0	1	0	undefined	1	undefined
30°	12	√32	1√3	2	2√3	√3
45°	1√2	1√2	1	√2	√2	1
60°	√32	12	√3	2√3	2	1√3
90°	1	0	undefined	1	undefined	0

Key rule: A final answer with a surd in the denominator (like 1√2 or 32+2√3) is not fully simplified. Always multiply the top and bottom by the conjugate to clear the surd. This is the most tested step in Q1 parts (iii) and (iv).

How to Evaluate Expressions at Specific Angles in Exercise 8.2 Question 1

Q1 has five parts. The method is the same for every part: look up the values from the table, substitute them, then simplify (and rationalise if needed).

Part	Expression	Key step	Answer
(i)	sin 60° cos 30° + sin 30° cos 60°	Substitute and add: √32 × √32 + 12 × 12 = 34 + 14	1
(ii)	2 tan² 45° + cos² 30° − sin² 60°	2(1)² + 34 − 34 = 2	2
(iii)	cos 45°sec 30° + cosec 30°	Substitute then rationalise by √2(√3−1)	3√2 − √68
(iv)	sin 30° + tan 45° − cosec 60°sec 30° + cos 60° + cot 45°	Reduce to 3√3 − 43√3 + 4, rationalise	43 − 24√311
(v)	5 cos² 60° + 4 sec² 30° − tan² 45°sin² 30° + cos² 30°	Denominator = 1 by Pythagorean identity; numerator: 54 + 163 − 1 = 6712	6712

In part (v), spot that the denominator is sin² 30° + cos² 30° = 1 before computing anything. The whole fraction collapses to just the numerator, saving one division step.

Exercise 8.2 Previous Year Questions and CBSE Board Weightage

Exercise 8.2 concepts appear in the CBSE Class 10 board paper regularly. Students who prepare this exercise can expect 2 to 3 marks from direct substitution or true/false reasoning questions.

Year	Question type from Exercise 8.2	Marks
2025	Evaluate expression using specific angle values (similar to Q1)	2
2024	True/false with justification on behaviour of sin and cos	2
2023	Find angles A and B from tangent equations (similar to Q3)	3
2022	MCQ on identifying which trig ratio matches a given expression	1
2021	Evaluate expression with rationalisation at specific angles	2

Students who score full marks on Exercise 8.2 board questions consistently do two things: check the denominator for a Pythagorean identity before computing, and rationalise every surd from the denominator as the last step before writing the final answer.

Other Resources for Chapter 8 Introduction to Trigonometry Class 10 Maths

Use the table below to move between the other resources for Chapter 8 and the other exercises of Introduction to Trigonometry.

Resource	Open page
Full chapter solutions	Introduction to Trigonometry Class 10 NCERT Solutions
Previous exercise (Exercise 8.1)	Introduction to Trigonometry Class 10 NCERT Solutions Exercise 8.1
Next exercise (Exercise 8.3)	Introduction to Trigonometry Class 10 NCERT Solutions Exercise 8.3
Revision notes	Introduction to Trigonometry Class 10 Notes
Formula sheet	Introduction to Trigonometry Class 10 Formula Sheet
NCERT book PDF	Introduction to Trigonometry Class 10 NCERT Book PDF
Handwritten notes	Introduction to Trigonometry Class 10 Handwritten Notes
Exemplar solutions	Introduction to Trigonometry Class 10 NCERT Exemplar Solutions
NCERT Exemplar PDF	Introduction to Trigonometry Class 10 Exemplar Book PDF
Previous year questions	Introduction to Trigonometry Class 10 PYQ

NCERT Solutions for Class 10 Maths Introduction to Trigonometry: All Exercises

Chapter 8 has three exercises. The table below links each exercise to its own step-by-step solution page.

Exercise	Solutions page
Exercise 8.1	Introduction to Trigonometry Exercise 8.1 Solutions
Exercise 8.2 (this page)	Introduction to Trigonometry Exercise 8.2 Solutions
Exercise 8.3	Introduction to Trigonometry Exercise 8.3 Solutions
Full chapter	Introduction to Trigonometry NCERT Solutions (all exercises)

NCERT Solutions for Class 10 Maths: All Chapters

Move between chapters using the table below. Each link opens that chapter's NCERT Solutions page.

Chapter	NCERT Solutions
Chapter 1	Real Numbers NCERT Solutions
Chapter 2	Polynomials NCERT Solutions
Chapter 3	Pair of Linear Equations NCERT Solutions
Chapter 4	Quadratic Equations NCERT Solutions
Chapter 5	Arithmetic Progressions NCERT Solutions
Chapter 6	Triangles NCERT Solutions
Chapter 7	Coordinate Geometry NCERT Solutions
Chapter 9	Some Applications of Trigonometry NCERT Solutions
Chapter 10	Circles NCERT Solutions
Chapter 12	Surface Areas and Volumes NCERT Solutions
Chapter 13	Statistics NCERT Solutions
Chapter 14	Probability NCERT Solutions

All NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8.2 with Step-by-Step Solutions

Exercise 8.2

Q 8.1

Evaluate the following:
(i) 60^∘30^∘+30^∘60^∘
(ii) 2tan² 45^∘+cos² 30^∘-sin² 60^∘
(iii) 45^∘30^∘+30^∘
(iv) 30^∘+45^∘-60^∘30^∘+60^∘+45^∘
(v) 5cos² 60^∘+4sec² 30^∘-tan² 45^∘sin² 30^∘+cos² 30^∘.

Q 8.2

Choose the correct option and justify your choice:
(i) 230^∘1+tan² 30^∘=   (A) 60^∘ (B) 60^∘ (C) 60^∘ (D) 30^∘
(ii) 1-tan² 45^∘1+tan² 45^∘=   (A) 90^∘ (B) 1 (C) 45^∘ (D) 0
(iii) 2A=2sin A is true when A=   (A) 0^∘ (B) 30^∘ (C) 45^∘ (D) 60^∘
(iv) 230^∘1-tan² 30^∘=   (A) 60^∘ (B) 60^∘ (C) 60^∘ (D) 30^∘.

Q 8.3

If tan(A+B)=3 and tan(A-B)=13; 0^∘∘; A>B, find A and B.

Q 8.4

State whether the following are true or false. Justify your answer.
(i) sin(A+B)=sin A+sin B.
(ii) The value of sinθ increases as θ increases.
(iii) The value of cosθ increases as θ increases.
(iv) sinθ=cosθ for all values of θ.
(v) cot A is not defined for A=0^∘.

Student Feedback

Out of 12,600 students surveyed before the 2026 boards, 82% said Q1 and Q2 were the most straightforward once they memorised the standard table, but rationalising surds in Q1 parts (iii) and (iv) was where most marks were lost. 4 out of 5 students who scored full marks always cleared surds from the denominator before writing the final answer.

Source: Collegedunia Class 10 student survey, 2026 board batch.

Introduction to Trigonometry Class 10 Maths Exercise 8.2 NCERT Solutions FAQs

Ques. How many questions are there in Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8.2?

Ans. Exercise 8.2 has 4 questions. Q1 has 5 sub-parts asking students to evaluate trigonometric expressions at specific angles (30°, 45°, 60°). Q2 is a 4-part MCQ where students choose the correct trig ratio and justify using substitution. Q3 asks students to find angles A and B from two tangent equations. Q4 has 5 true/false statements about the behaviour of sine, cosine, and cotangent. All questions are based on the 2026-27 CBSE syllabus.

Ques. What is the standard value table used in Exercise 8.2?

Ans. The standard table gives the exact values of all six trigonometric ratios at 0°, 30°, 45°, 60°, and 90°. The key values used in Exercise 8.2 are: sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, cos 60° = 1/2, tan 45° = 1, sec 30° = 2/√3, and cosec 30° = 2. Every part of Q1 and Q2 is solved by direct substitution from this table.

Ques. Why is rationalisation needed in Exercise 8.2 Q1 parts (iii) and (iv)?

Ans. After substituting the standard angle values in parts (iii) and (iv), the denominator contains surds such as √2, √3, or expressions like 2 + 2√3. A fraction with a surd in the denominator is not in its simplest form. Students must multiply the numerator and denominator by the conjugate (or the same surd) to clear the root from the denominator. The board examiner expects a surd-free denominator in the final answer, so leaving surds below the line costs marks.

Ques. How do you solve Q3 of Exercise 8.2 to find A and B?

Ans. Q3 gives tan(A + B) = √3 and tan(A − B) = 1/√3. From the standard table, tan 60° = √3 and tan 30° = 1/√3. Since 0° < A + B ≤ 90°, we get A + B = 60°. Since A > B, A − B = 30°. Adding both equations: 2A = 90°, so A = 45°. Subtracting: 2B = 30°, so B = 15°. Both values satisfy the given constraints.

Ques. Are these Exercise 8.2 solutions based on the 2026-27 CBSE syllabus?

Ans. Yes. All solutions follow the current 2026-27 syllabus for Class 10 Mathematics. Exercise 8.2 on specific angle values and the behaviour of trigonometric ratios is fully retained in the latest NCERT edition. The questions in this exercise are directly tested in the CBSE Class 10 board paper, typically carrying 2 to 3 marks each.