Question:
Evaluate : \( \tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right) \).
Evaluate : \( \tan \left( \sin^{-1} 1 - \cos^{-1} \left( -\frac{1}{2} \right) \right) \).
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Remember that \( \cos^{-1} \) of a negative value always lies in the second quadrant \( (\frac{\pi}{2}, \pi] \).
\( \tan(-\theta) = -\tan \theta \), a property of odd functions.
\( \tan(-\theta) = -\tan \theta \), a property of odd functions.
Updated On: Sep 10, 2026
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Solution and Explanation
Concept:
• Principal values:
• \( \sin^{-1}(1) = \frac{\pi}{2} \) since \( \sin \frac{\pi}{2} = 1 \).
• \( \cos^{-1}(-x) = \pi - \cos^{-1}(x) \).
Step 1: Determine the principal values of inverse terms
For the first term: \[ \sin^{-1}(1) = \frac{\pi}{2} \] For the second term: \[ \cos^{-1} \left( -\frac{1}{2} \right) = \pi - \cos^{-1} \left( \frac{1}{2} \right) \] Since \( \cos \frac{\pi}{3} = \frac{1}{2} \): \[ \cos^{-1} \left( -\frac{1}{2} \right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]
Step 2: Substitute values into the expression
The expression becomes: \[ \tan \left( \frac{\pi}{2} - \frac{2\pi}{3} \right) \]
Step 3: Simplify and calculate the final value
Find the common denominator for the angle: \[ \frac{\pi}{2} - \frac{2\pi}{3} = \frac{3\pi - 4\pi}{6} = -\frac{\pi}{6} \] Now find the tangent: \[ \tan \left( -\frac{\pi}{6} \right) = -\tan \left( \frac{\pi}{6} \right) \] \[ = -\frac{1}{\sqrt{3}} \]
• Principal values:
• \( \sin^{-1}(1) = \frac{\pi}{2} \) since \( \sin \frac{\pi}{2} = 1 \).
• \( \cos^{-1}(-x) = \pi - \cos^{-1}(x) \).
Step 1: Determine the principal values of inverse terms
For the first term: \[ \sin^{-1}(1) = \frac{\pi}{2} \] For the second term: \[ \cos^{-1} \left( -\frac{1}{2} \right) = \pi - \cos^{-1} \left( \frac{1}{2} \right) \] Since \( \cos \frac{\pi}{3} = \frac{1}{2} \): \[ \cos^{-1} \left( -\frac{1}{2} \right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]
Step 2: Substitute values into the expression
The expression becomes: \[ \tan \left( \frac{\pi}{2} - \frac{2\pi}{3} \right) \]
Step 3: Simplify and calculate the final value
Find the common denominator for the angle: \[ \frac{\pi}{2} - \frac{2\pi}{3} = \frac{3\pi - 4\pi}{6} = -\frac{\pi}{6} \] Now find the tangent: \[ \tan \left( -\frac{\pi}{6} \right) = -\tan \left( \frac{\pi}{6} \right) \] \[ = -\frac{1}{\sqrt{3}} \]
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