Prove that ( A - A)/( A + A) = (² A)/((1 + A)²).

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2023: question 31(a)

Set in the CBSE Class 10 board exam of 2023, paper 430/6/1, section C, question 31(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

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

Prove that $\frac{cot A - cos A}{cot A + cos A} = \frac{cos ^{2} A}{( 1 + sin A ) ^{2}}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/6/1.

Hints

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

Hint 1

Replace cot A by cos A / sin A.

Hint 2

After simplifying to (1 − sin A)/(1 + sin A), multiply top and bottom by (1 + sin A).

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 $= \frac{cot A - cos A}{cot A + cos A} = \frac{\frac{c o s A}{s i n A} - cos A}{\frac{c o s A}{s i n A} + cos A}$
½ mark
Taking $\frac{cos A}{sin A}$ common from numerator and denominator: $= \frac{1 - sin A}{1 + sin A}$
1 mark
Multiplying numerator and denominator by $(1 + sin A)$ : $= \frac{( 1 - sin A ) ( 1 + sin A )}{( 1 + sin A ) ( 1 + sin A )}$
1 mark
$= \frac{1 - sin ^{2} A}{( 1 + sin A ) ^{2}} = \frac{cos ^{2} A}{( 1 + sin A ) ^{2}} =$ RHS [since $sin^{2} A + cos^{2} A = 1$ ]
½ mark

Answer: LHS = RHS, hence proved

Where the marks are

Writes cot A as cos A / sin A (½ mark)
Simplifies to (1 − sin A)/(1 + sin A) (1 mark)
Multiplies by (1 + sin A) above and below (1 mark)
Uses identity to reach cos²A/(1 + sin A)² (½ 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 430/6/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 31(a)

Hints

Step-by-step solution

Revise the chapter

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