Question

If \(\tan \theta = \cot \theta\) then the value of \(sec\ θ \)=

• A. \(\frac{1}{\sqrt{2}}\)
• B. \(\frac{\sqrt{3}}{2}\)
• C. \(\sqrt{2}\)
• D. 1

Answer 1

The Correct Option is C

To solve the problem, we need to determine the value of \(\sec \theta\) given that \(\tan \theta = \cot \theta\).

1. Understanding the Relationship Between \(\tan \theta\) and \(\cot \theta\):
By definition, \(\cot \theta = \frac{1}{\tan \theta}\). Therefore, the equation \(\tan \theta = \cot \theta\) can be rewritten as: \[ \tan \theta = \frac{1}{\tan \theta} \] Multiplying both sides by \(\tan \theta\) (assuming \(\tan \theta \neq 0\)): \[ \tan^2 \theta = 1 \] Taking the square root of both sides: \[ \tan \theta = \pm 1 \]

2. Determining \(\sec \theta\) for \(\tan \theta = \pm 1\):
The secant function is defined as \(\sec \theta = \frac{1}{\cos \theta}\). To find \(\sec \theta\), we first need to determine \(\cos \theta\) when \(\tan \theta = \pm 1\).

Case 1: \(\tan \theta = 1\)
If \(\tan \theta = 1\), then \(\theta = 45^\circ\) (or \(\frac{\pi}{4}\) radians). For \(\theta = 45^\circ\): \[ \cos \theta = \cos 45^\circ = \frac{1}{\sqrt{2}} \] Thus: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

Case 2: \(\tan \theta = -1\)
If \(\tan \theta = -1\), then \(\theta = 135^\circ\) (or \(\frac{3\pi}{4}\) radians). For \(\theta = 135^\circ\): \[ \cos \theta = \cos 135^\circ = -\frac{1}{\sqrt{2}} \] Thus: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{1}{\sqrt{2}}} = -\sqrt{2} \] However, since the problem does not specify the quadrant, we generally consider the positive value of \(\sec \theta\) unless otherwise stated.

3. Conclusion:
From both cases, the value of \(\sec \theta\) is \(\sqrt{2}\).

Final Answer:
The value of \(\sec \theta\) is \({\sqrt{2}}\).