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

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/5/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2023: question 25(a)

Set in the CBSE Class 10 board exam of 2023, paper 30/5/1, section B, question 25(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25(a) · 2 marks · Very short answerIntroduction to Trigonometry

If $a cos θ + b sin θ = m$ and $a sin θ - b cos θ = n$ , then prove that $a^{2} + b^{2} = m^{2} + n^{2}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/5/1.

Hints

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

Hint 1

Start from $m^{2} + n^{2}$ and substitute the given expressions.

Hint 2

After expanding, notice that the middle terms cancel; then group the $a^{2}$ terms and the $b^{2}$ terms.

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.

$m^{2} + n^{2} = (a cos θ + b sin θ)^{2} + (a sin θ - b cos θ)^{2}$
½ mark
$= a^{2} cos^{2} θ + b^{2} sin^{2} θ + 2 ab sin θ cos θ + a^{2} sin^{2} θ + b^{2} cos^{2} θ - 2 ab sin θ cos θ$
$= a^{2} (cos^{2} θ + sin^{2} θ) + b^{2} (sin^{2} θ + cos^{2} θ)$
1 mark
$= a^{2} + b^{2}$ [since $sin^{2} θ + cos^{2} θ = 1$ ]. Hence $a^{2} + b^{2} = m^{2} + n^{2}$ .
½ mark

Answer: m² + n² = a² + b² (proved).

Where the marks are

Writes m² + n² as the sum of the two squared expressions (½ mark)
Expands, cancels the 2ab sinθ cosθ terms and groups as a²(cos²θ + sin²θ) + b²(sin²θ + cos²θ) (1 mark)
Uses sin²θ + cos²θ = 1 to get a² + b² (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 paper2023 board paper, set 30/5/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 25(a)

Hints

Step-by-step solution

Revise the chapter

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