Prove that: (cosec A - sin A)(sec A - cos A) = (1)/(tan A + cot A).

Both sides equal A A; hence proved.

MathsCBSE Class 10 · Basic · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic sample paper 2024-25: question 28

From the official CBSE sample question paper for 2024-25, section C, question 28, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28 · 3 marks · Short answerIntroduction to Trigonometry

Prove that: $(cosec A - s in A) (sec A - cos A) = \frac{1}{t an A + co t A}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Write every ratio in terms of $sin A$ and $cos A$ .

Hint 2

Simplify each side separately. Both reduce to the same simple product.

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 $= (\frac{1}{s i n A} - sin A) (\frac{1}{c o s A} - cos A)$
½ mark
$= \frac{1 - s i n ^{2} A}{s i n A} \times \frac{1 - c o s ^{2} A}{c o s A} = \frac{c o s ^{2} A}{s i n A} \times \frac{s i n ^{2} A}{c o s A}$ [using $sin^{2} A + cos^{2} A = 1$ ]
1 mark
$= sin A cos A$
½ mark
RHS $= \frac{1}{\frac{s i n A}{c o s A} + \frac{c o s A}{s i n A}} = \frac{s i n A c o s A}{s i n ^{2} A + c o s ^{2} A} = sin A cos A$ .
Hence LHS = RHS. Proved.
1 mark

Answer: Both sides equal $sin A cos A$ ; hence proved.

Where the marks are

Writes cosec A and sec A in terms of sin A and cos A (½ mark)
Uses $1 - sin^{2} A = cos^{2} A$ and $1 - cos^{2} A = sin^{2} A$ (1 mark)
Simplifies LHS to $sin A cos A$ (½ mark)
Simplifies RHS to $sin A cos A$ (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 2024-25 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2024-25: question 28

Hints

Step-by-step solution

Revise the chapter

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