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Question

Mathematics Question on Trigonometric Equations

The minimum value of sec⁡2 θ+cosec2 θ\sec^2 \, \theta + cosec^2 \, \theta is

A

2

B

4

C

0

D

1

Answer

4

Explanation

Solution

sec⁡2θ+cosec2θ=1+tan⁡2θ+1+cot⁡2θ\sec^{2} \theta + cosec^{2} \theta = 1 + \tan^{2} \theta + 1 +\cot^{2} \theta =2+(tan⁡θ−cot⁡θ)2+2tan⁡θcot⁡θ= 2 + \left(\tan\theta - \cot\theta\right)^{2} + 2 \tan\theta \cot\theta =2+2+(tan⁡θ−cot⁡θ)2≥4.= 2 + 2 +\left(\tan\theta - \cot \theta\right)^{2} \ge4.