Prove that A(1 - A)( A + A) = 1.

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 29(B)

Set in the CBSE Class 10 board exam of 2023, paper 30/4/1, section C, question 29(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(B) · 3 marks · Short answerIntroduction to Trigonometry

Prove that $sec A (1 - sin A) (sec A + tan A) = 1$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/4/1.

Hints

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

Hint 1

Change sec and tan into sine and cosine.

Hint 2

You will get $(1 - sin A) (1 + sin A)$ . Which identity simplifies this?

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.

Write everything in terms of sine and cosine:
$LHS = \frac{1}{cos A} (1 - sin A) (\frac{1}{cos A} + \frac{sin A}{cos A})$
1 mark
Combine the bracket:
$= \frac{1}{cos A} (1 - sin A) \cdot \frac{1 + sin A}{cos A}$
1 mark
Multiply out:
$= \frac{1 - sin ^{2} A}{cos ^{2} A} = \frac{cos ^{2} A}{cos ^{2} A} = 1 = RHS$ [since $1 - sin^{2} A = cos^{2} A$ ]
Hence proved.
1 mark

Answer: LHS = 1 = RHS (proved).

Where the marks are

Converts $sec A$ and $tan A$ to $\frac{1}{c o s A}$ and $\frac{s i n A}{c o s A}$ (1 mark)
Simplifies the bracket to $\frac{1 + s i n A}{c o s A}$ (1 mark)
Uses $1 - sin^{2} A = cos^{2} A$ to get 1 (1 mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)

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 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 29(B)

Hints

Step-by-step solution

Revise the chapter

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