If (1)/( x - x) = cosec\ x√(2), then prove that ((1)/( x + x) )² = (² x)/(2).

Proved: ((1)/( x + x) )² = (² x)/(2)

MathsCBSE Class 10 · Standard · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2026-27: question 31(B)

From the official CBSE sample question paper for 2026-27, section C, question 31(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(B) · 3 marks · Short answerIntroduction to Trigonometry

If $\frac{1}{s i n x - c o s x} = \frac{cosec x}{2}$ , then prove that $(\frac{1}{s i n x + c o s x})^{2} = \frac{s e c ^{2} x}{2}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Write $cosec x$ as $\frac{1}{s i n x}$ and take reciprocals of both sides of the given condition.

Hint 2

Square $sin x - cos x = 2 sin x$ to find $2 sin x cos x$ in terms of $sin^{2} x$ . Then expand $(sin x + cos x)^{2}$ in the LHS and substitute.

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.

Given $\frac{1}{sin x - cos x} = \frac{cosec x}{2} = \frac{1}{2 sin x}$ .
Taking reciprocals: $sin x - cos x = 2 sin x$ .
½ mark
Squaring both sides:
$sin^{2} x + cos^{2} x - 2 sin x cos x = 2 sin^{2} x$
$\Rightarrow 1 - 2 sin x cos x = 2 sin^{2} x$
$\Rightarrow 2 sin x cos x = 1 - 2 sin^{2} x$ …(i)
1 mark
LHS $= (\frac{1}{sin x + cos x})^{2} = \frac{1}{sin ^{2} x + cos ^{2} x + 2 sin x cos x} = \frac{1}{1 + 2 sin x cos x}$ .
½ mark
Using (i): $= \frac{1}{1 + 1 - 2 sin ^{2} x} = \frac{1}{2 ( 1 - sin ^{2} x )} = \frac{1}{2 cos ^{2} x}$ .
½ mark
$= \frac{sec ^{2} x}{2} =$ RHS. Hence proved.
½ mark

Answer: Proved: $(\frac{1}{s i n x + c o s x})^{2} = \frac{s e c ^{2} x}{2}$

Where the marks are

Simplifies the given condition to $sin x - cos x = 2 sin x$ (½ mark)
Squares to get $2 sin x cos x = 1 - 2 sin^{2} x$ (1 mark)
Writes LHS as $\frac{1}{1 + 2 s i n x c o s x}$ (½ mark)
Substitutes to get $\frac{1}{2 c o s ^{2} x}$ (½ mark)
Concludes $\frac{s e c ^{2} x}{2}$ = RHS (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 2026-27 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2026-27: question 31(B)

Hints

Step-by-step solution

Revise the chapter

More questions from Introduction to Trigonometry