Question

If \[ \cos\theta \cosec\theta=-1 \] and \(\theta\) lies in the second quadrant, then \(\cos\theta=\)

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

Hint:

In the second quadrant, \(\sin\theta\) is positive but \(\cos\theta\) and \(\tan\theta\) are negative.

Answer 1

The Correct Option is C

Concept:
We know: \[ \cosec\theta=\frac{1}{\sin\theta} \] Therefore: \[ \cos\theta\cosec\theta=\frac{\cos\theta}{\sin\theta}=\cot\theta \]

Step 1: Given: \[ \cos\theta\cosec\theta=-1 \]

Step 2: Convert into cotangent: \[ \cot\theta=-1 \]

Step 3: Therefore: \[ \tan\theta=-1 \]

Step 4: Since \(\theta\) lies in the second quadrant, sine is positive and cosine is negative.

Step 5: The reference angle for \(|\tan\theta|=1\) is: \[ 45^\circ \]

Step 6: In the second quadrant: \[ \cos\theta=-\frac{1}{\sqrt{2}} \] \[ \cos\theta=-\frac{\sqrt{2}}{2} \] \[ \boxed{-\frac{\sqrt{2}}{2}} \]