Given that θ + θ = x, prove that ^4θ + ^4θ = (2 - (x²-1)²)/(2).

^4θ + ^4θ = (2-(x²-1)²)/(2) (proved)

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 30(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, section C, question 30(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(b) · 3 marks · Short answerIntroduction to Trigonometry

Given that $sin θ + cos θ = x$ , prove that $sin^{4} θ + cos^{4} θ = \frac{2 - ( x ^{2} - 1 ) ^{2}}{2}$ .

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 the given relation to find $x^{2} - 1$ in terms of $sin θ cos θ$ .

Hint 2

Start from RHS and use $1 = (sin^{2} θ + cos^{2} θ)^{2}$ .

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.

Squaring $sin θ + cos θ = x$ : $sin^{2} θ + cos^{2} θ + 2 sin θ cos θ = x^{2} \Rightarrow 2 sin θ cos θ = x^{2} - 1$
1 mark
RHS $= \frac{2 - ( 2 sin θ cos θ ) ^{2}}{2} = \frac{2 - 4 sin ^{2} θ cos ^{2} θ}{2}$
½ mark
$= 1 - 2 sin^{2} θ cos^{2} θ$
½ mark
$= (sin^{2} θ + cos^{2} θ)^{2} - 2 sin^{2} θ cos^{2} θ$
½ mark
$= sin^{4} θ + cos^{4} θ =$ LHS. Hence proved.
½ mark

Answer: $sin^{4} θ + cos^{4} θ = \frac{2 - ( x ^{2} - 1 ) ^{2}}{2}$ (proved)

Where the marks are

Obtains $2 sin θ cos θ = x^{2} - 1$ (1 mark)
Substitutes in RHS (½ mark)
Simplifies to $1 - 2 sin^{2} θ cos^{2} θ$ (½ mark)
Writes 1 as $(sin^{2} θ + cos^{2} θ)^{2}$ (½ mark)
Reaches LHS (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 30(b)

Hints

Step-by-step solution

Revise the chapter

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