Prove that : ( x - x + 1) · ( x - x) = ( x + x - 1)

MathsCBSE Class 10 · Basic · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic sample paper 2025-26: question 30

From the official CBSE sample question paper for 2025-26, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerIntroduction to Trigonometry

Prove that : $(sin x - cos x + 1) \cdot (sec x - tan x) = (sin x + cos x - 1)$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Write $sec x - tan x$ as a single fraction in $sin x$ and $cos x$ .

Hint 2

Group $(1 + sin x)$ and $- cos x$ separately; $(1 + sin x) (1 - sin x)$ gives an identity.

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.

LHS $= (sin x - cos x + 1) (\frac{1}{c o s x} - \frac{s i n x}{c o s x}) = (sin x - cos x + 1) \cdot \frac{1 - s i n x}{c o s x}$
1 mark
$= (1 + sin x) \frac{1 - s i n x}{c o s x} - cos x \cdot \frac{1 - s i n x}{c o s x} = \frac{1 - s i n ^{2} x}{c o s x} - (1 - sin x)$
1 mark
$= \frac{c o s ^{2} x}{c o s x} - 1 + sin x = sin x + cos x - 1 =$ RHS. Hence proved.
1 mark

Answer: LHS = $sin x + cos x - 1$ = RHS (proved)

Where the marks are

Writes $sec x - tan x = \frac{1 - s i n x}{c o s x}$ (1 mark)
Splits and simplifies to $\frac{1 - s i n ^{2} x}{c o s x} - (1 - sin x)$ (1 mark)
Uses $1 - sin^{2} x = cos^{2} x$ to reach RHS (1 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 2025-26 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2025-26: question 30

Hints

Step-by-step solution

Revise the chapter

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