2026/27 Enrolment Open · Free Assessment

Chapter 8 · Case-study practice

Introduction to Trigonometry: Case Study PYQs with Answers

A trigonometry case study asks you to choose the correct ratios and simplify expressions using identities. In the current CBSE sample-paper format, Section E has three case studies worth 4 marks each. Practise this selected board case on a right triangle, with all parts and both OR alternatives solved in NCERT style.

1 distinct case · 1 board-paper appearance · Basic accessible paper
Solutions prepared by Team TutorVertex
Verified by Siddharth Sir
Last updated: 27 October 2026

Which relationship?

What is asked?Use
Missing hypotenuseAC² = AB² + BC², when ∠B = 90°
Ratios for angle Asin A = BCAC, cos A = ABAC, tan A = BCAB
Reciprocal expressionssec A = 1cos A, cosec A = 1sin A, cot A = 1tan A
Simplify squaressin² A + cos² A = 1
sec² A − tan² A = 1
cosec² A − cot² A = 1
Basic4 marks · Accessible paper

Case 1. A right triangle and trigonometric identities

A teacher asked his students to draw a right triangle ABC with AB = 8 cm, ∠B = 90° and BC = 15 cm. Based on the above, answer the following:

Appeared in: 2025 · Basic · Main

(i) Evaluate (sin² A − cos² A). [1 mark]

View solution — (i)
Right triangle ABCA is at the bottom left, B at the bottom right, and C above B. AB is 8 cm, BC is 15 cm and the right angle is at B. Angle A is marked. The calculated hypotenuse AC is 17 cm.ABC8 cm15 cm17 cmA
TutorVertex solution diagram, drawn from the given measurements; not to scale.

In △ABC, ∠B = 90°.

By Pythagoras theorem, AC² = AB² + BC².

AC² = 8² + 15² = 289.

AC = 17 cm.

sin A = BCAC = 1517, and cos A = ABAC = 817.

sin² A − cos² A = (15/17)² − (8/17)².

= 225 − 64289 = 161289.

Answer: 161289

Answer matched to the official CBSE marking scheme. Working prepared by Team TutorVertex.

(ii) Evaluate (1cos² A − 1cot² A). [1 mark]

View solution — (ii)

1cos² A − 1cot² A = sec² A − tan² A.

Using sec² A = 1 + tan² A,

sec² A − tan² A = 1.

Answer: 1

Answer matched to the official CBSE marking scheme. Working prepared by Team TutorVertex.

(iii)(a) Evaluate 2 tan A1 + tan² A and prove that it is equal to 2 sin A cos A. [2 marks]

View solution — (iii)(a)

From the given triangle, sin A = 1517, cos A = 817 and tan A = 158.

Using 1 + tan² A = sec² A,

2 tan A1 + tan² A = 2 tan Asec² A.

= 2 tan A × cos² A.

= 2 × sin Acos A × cos² A.

= 2 sin A cos A.

2 sin A cos A = 2 × 1517 × 817 = 240289.

Hence, both expressions have the value 240289.

Answer: 240289; both expressions are equal

Answer matched to the official CBSE marking scheme. Working prepared by Team TutorVertex.

OR

(iii)(b) Evaluate tan² A − sec² Acot² A − cosec² A. [2 marks]

View solution — (iii)(b)

Using sec² A = 1 + tan² A,

tan² A − sec² A = −1.

Using cosec² A = 1 + cot² A,

cot² A − cosec² A = −1.

tan² A − sec² Acot² A − cosec² A = −1−1 = 1.

Answer: 1

Answer matched to the official CBSE marking scheme. Working prepared by Team TutorVertex.

Questions students ask

How do I choose the sides for sin A and cos A?

Use the named angle A. In this triangle, BC is opposite A, AB is adjacent to A, and AC is the hypotenuse. Hence sin A = BC/AC and cos A = AB/AC.

Why does 1/cos² A − 1/cot² A equal 1?

The reciprocal terms are sec² A and tan² A. The identity sec² A − tan² A = 1 gives the result directly.

Why is 1/cot² A the same as tan² A?

Since cot A = 1/tan A, squaring gives cot² A = 1/tan² A. Taking the reciprocal gives 1/cot² A = tan² A.

How do I prove an expression equals 2 sin A cos A?

Start with the expression you need to simplify. Here, replace 1 + tan² A by sec² A, then use tan A = sin A/cos A and 1/sec² A = cos² A.

Should I answer both OR alternatives?

Answer one alternative where an OR is printed. Both alternatives are solved here for practice; follow the marks beside each part.

Answer summary

Open the answer summary
Case / partAnswer
1 (i)161289
1 (ii)1
1 (iii)(a)240289; both expressions are equal
1 (iii)(b)1

Both OR alternatives are listed. Answer one where an OR is given in the examination.

How to approach a case study

  1. Read the full case and identify the right angle and the angle named in each part.
  2. Find a missing side only when needed, then write the relevant ratios.
  3. Substitute exact values or use a suitable identity, with one calculation per line.
  4. Check the expression and give a clear final answer. Attempt one alternative wherever an OR is printed.

About these questions and solutions

This selected CBSE case retains its passage, mathematical expressions, marks and OR choice.

Solutions prepared by Team TutorVertex, using NCERT-style working and answers matched to the corresponding CBSE marking scheme.

Official sources — View paper and marking scheme

Case 1 source

2025 main · Basic · 430(B) · Q37 · Accessible paper

Left out and why: adjacent plots

The 2025 Basic adjacent-plots case contains incompatible measurements in its diagram. The lower triangle gives AG = 38 m, so AD = AG + GD = 78 m. The upper right triangle gives CD = DB = HD + HB = 50 m; its marked 60° angle would then require AD = 50√3 m, which is not 78 m.

The corresponding schemes use AD = 78 m and print sin 60° = 78/AC, followed by AC = 89.96 m. These lines do not resolve the conflicting values of AD.

Because these conditions cannot all hold together, this complete case is left out of the solved selection.

Papers: 430/4/1 Q38, page 15; 430/4/2 Q36, page 13; 430/4/3 Q37, page 15. Corresponding scheme solutions are on pages 18, 16 and 18. The diagram and scheme pages were checked visually.

Official question papers · Official marking schemes

Official 2026–27 Standard sample paper: Section E format

NCERT Class 10 Mathematics: Chapter 8, Introduction to Trigonometry

CBSE Class 10 Mathematics syllabus, 2026–27

Need help with trigonometric identities?

Connect with TutorVertex for personalised 1-to-1 Maths tutoring.

Book a Free Assessment →