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NDA 2 2025 Maths, question 46

Topic: Trigonometry

Q46

What is cot⁡−19+cosec⁡−1(414)\cot ^{-1}9+\operatorname{cosec} ^{-1}\left( \frac{\sqrt{41}}{4} \right) equal to?

  1. (a) π4\frac{\pi}{4}Correct answer
  2. (b) π3\frac{\pi}{3}
  3. (c) π2\frac{\pi}{2}
  4. (d) π\pi

Solution.

Answer: (a) π4\frac{\pi}{4}

  1. Let A=cot⁻¹9 and B=cosec⁻¹(√41/4).
  2. Both are acute.
  3. Then tan A=1/9, sin B=4/√41 and cos B=5/√41, so tan B=4/5.
  4. Thus tan(A+B)=(1/9+4/5)/(1−4/45)=1.
  5. Since the sum is acute, A+B=π/4.