Question

The value of \(\cos(\sec^{-1}x+\text{cosec}^{-1}x)\), \(|x|\ge 1\) is

• A. 1
• B. 0
• C. -1
• D. \(\dfrac{1}{2}\)

Hint:

sec⁻¹x + cosec⁻¹x is always π/2 for |x| ≥ 1.

Answer 1

The Correct Option is B

Step 1: Testing a convenient value:
Take \(x=1\): \(\sec^{-1}1=0\), \(\text{cosec}^{-1}1=\dfrac{\pi}{2}\).

Step 2: Computing:
Sum \(=0+\dfrac{\pi}{2}=\dfrac{\pi}{2}\), so \(\cos\left(\dfrac{\pi}{2}\right)=0\).

Step 3: Generalising:
Since the identity \(\sec^{-1}x+\text{cosec}^{-1}x=\pi/2\) holds for all valid \(x\), the value is always \(0\), not just at \(x=1\).

Final Answer:
\[ \boxed{0} \]