Question

If \[ \sin^{-1}x+\cos^{-1}x=\theta \] find \(\theta\).

• A. \(0\)
• B. \(\pi\)
• C. \(\pi/2\)
• D. \(2\pi\)

Hint:

Memorize the identity \(\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\); it is frequently asked in CUET examinations.

Answer 1

The Correct Option is C

Concept: One of the most important inverse trigonometric identities is \[ \sin^{-1}x+\cos^{-1}x = \frac{\pi}{2} \] for every \(x\in[-1,1]\).

Step 1: Apply the standard identity Given \[ \sin^{-1}x+\cos^{-1}x=\theta \] Using the identity, \[ \theta=\frac{\pi}{2} \]

Step 2: Verify with an example Take \[ x=\frac12 \] Then \[ \sin^{-1}\left(\frac12\right)=\frac{\pi}{6} \] and \[ \cos^{-1}\left(\frac12\right)=\frac{\pi}{3} \] Therefore, \[ \frac{\pi}{6}+\frac{\pi}{3} = \frac{\pi}{2} \] which confirms the identity. Final Answer: \[ \boxed{\frac{\pi}{2}} \]