Exams › SSC CGL (Prelims) › General

If \(\tan\theta + \cot\theta = 2\) and \(\theta \in (0,\pi/2)\), then what is \(\sec\theta + \csc\theta\)?

1. 2 \sqrt 2
2. \sqrt 2
3. 2
4. 4 \sqrt 2

Correct answer: 2 \sqrt 2

Solution

Given \(\tan\theta + \cot\theta = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \frac{1}{\sin\theta\cos\theta} = 2\), so \(\sin\theta\cos\theta = \frac12\). Now \((\sec\theta + \csc\theta)^2 = \sec^2\theta + \csc^2\theta + 2\sec\theta\csc\theta = \frac{1}{\cos^2\theta} + \frac{1}{\sin^2\theta} + \frac{2}{\sin\theta\cos\theta}\). This simplifies to \(\frac{1}{\sin^2\theta\cos^2\theta} = 8\), hence \(\sec\theta + \csc\theta = 2\sqrt2\).

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