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What is the number of solutions of the equation cot2x cot3x = 1 for 0 < x

This question was previously asked in
NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. Only one
  2. Only two
  3. Only five
  4. More than five

Answer (Detailed Solution Below)

Option 3 : Only five
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Detailed Solution

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Explanation:

Given

cot2x. cot3x = 1

⇒ cos2x. cos3x – sin2x.sin3x= 0

⇒ cos(2x+3x) = cos5x = 0    0 < x < π 

Hence 5x = \(\frac{\pi }{2}, \frac{3\pi }{2}, \frac{5\pi }{2}, \frac{7\pi }{2}, \frac{9\pi }{2}\)

⇒ x = \(\frac{\pi }{10}, \frac{3\pi }{10}, \frac{5\pi }{10}, \frac{7\pi }{10}, \frac{9\pi }{10}\)

Thus, only five solutions are possible

∴ the correct answer is option C

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