If θ + θ = 1, then prove that θ - θ = ± 1.

MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 29

From the official CBSE sample question paper for 2024-25, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerIntroduction to Trigonometry

If $cos θ + sin θ = 1$ , then prove that $cos θ - sin θ = \pm 1$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Expand $(cos θ + sin θ)^{2}$ and $(cos θ - sin θ)^{2}$ . What happens when you add them?

Hint 2

The sum simplifies to 2. Substitute the given value and solve for $(cos θ - sin θ)^{2}$ .

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.

Expanding and adding: $(cos θ + sin θ)^{2} + (cos θ - sin θ)^{2} = 2 (cos^{2} θ + sin^{2} θ) = 2$ [since $sin^{2} θ + cos^{2} θ = 1$ ].
1½ marks
Substituting $cos θ + sin θ = 1$ : $1^{2} + (cos θ - sin θ)^{2} = 2 \Rightarrow (cos θ - sin θ)^{2} = 1$ .
1 mark
Taking square roots: $cos θ - sin θ = \pm 1$ . Hence proved.
½ mark

Answer: $cos θ - sin θ = \pm 1$ (proved)

Where the marks are

Uses $(cos θ + sin θ)^{2} + (cos θ - sin θ)^{2} = 2$ through the identity (1½ marks)
Substitutes the given value to get $(cos θ - sin θ)^{2} = 1$ (1 mark)
Concludes $cos θ - sin θ = \pm 1$ (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 paperSample paper 2024-25 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2024-25: question 29

Hints

Step-by-step solution

Revise the chapter

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