If aθ + bθ = m and bθ + aθ = n, prove that a² + n² = b² + m².

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2025: question 22(a)

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, section B, question 22(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22(a) · 2 marks · Very short answerIntroduction to Trigonometry

If $a sec θ + b tan θ = m$ and $b sec θ + a tan θ = n$ , prove that $a^{2} + n^{2} = b^{2} + m^{2}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/6/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Square both given equations.

Hint 2

The $2 ab sec θ tan θ$ terms are identical — subtract to remove them and use $sec^{2} θ - tan^{2} θ = 1$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Squaring the first relation: $m^{2} = a^{2} sec^{2} θ + b^{2} tan^{2} θ + 2 ab sec θ tan θ$
½ mark
Squaring the second relation: $n^{2} = b^{2} sec^{2} θ + a^{2} tan^{2} θ + 2 ab sec θ tan θ$
½ mark
Subtracting: $m^{2} - n^{2} = a^{2} (sec^{2} θ - tan^{2} θ) - b^{2} (sec^{2} θ - tan^{2} θ)$
½ mark
Using $sec^{2} θ - tan^{2} θ = 1$ : $m^{2} - n^{2} = a^{2} - b^{2} \Rightarrow a^{2} + n^{2} = b^{2} + m^{2}$ . Hence proved.
½ mark

Answer: $a^{2} + n^{2} = b^{2} + m^{2}$ (proved)

Where the marks are

Correct expansion of $m^{2}$ (½ mark)
Correct expansion of $n^{2}$ (½ mark)
Subtracts and groups terms as $(sec^{2} θ - tan^{2} θ)$ (½ mark)
Uses identity to reach the result (½ mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Proving Simple Trigonometric Identities

Practise this chapterBoard-style questions from Introduction to Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 22(a)

Hints

Step-by-step solution

Revise the chapter

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