Question:
Evaluate : \(\sin\left(\tan^{-1}(-\sqrt{3}) - \sec^{-1} 2\right)\)
Evaluate : \(\sin\left(\tan^{-1}(-\sqrt{3}) - \sec^{-1} 2\right)\)
Show Hint
tan^{-1}(-root 3) = -pi/3 and sec^{-1}2 = pi/3, so the expression is sin(-2pi/3).
Updated On: Oct 22, 2026
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Solution and Explanation
Step 1: Understanding the Concept:
We find the principal value of each inverse function first. Then we evaluate the sine of the resulting angle.
Step 2: Find tan inverse of minus root 3.
The principal range of \(\tan^{-1}\) is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Since \(\tan\frac{\pi}{3} = \sqrt{3}\) and tangent is an odd function, \(\tan^{-1}(-\sqrt{3}) = -\dfrac{\pi}{3}\).
Step 3: Find sec inverse of 2.
The principal range of \(\sec^{-1}\) is \([0, \pi] - \left\{\frac{\pi}{2}\right\}\).
Since \(\sec\frac{\pi}{3} = 2\), we get \(\sec^{-1} 2 = \dfrac{\pi}{3}\).
Step 4: Substitute and evaluate.
\[ \sin\left(-\frac{\pi}{3} - \frac{\pi}{3}\right) = \sin\left(-\frac{2\pi}{3}\right) = -\sin\frac{2\pi}{3} \]
Now \(\sin\frac{2\pi}{3} = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
Final Answer:
The value is \(-\dfrac{\sqrt{3}}{2}\). \[ \boxed{-\frac{\sqrt{3}}{2}} \]
We find the principal value of each inverse function first. Then we evaluate the sine of the resulting angle.
Step 2: Find tan inverse of minus root 3.
The principal range of \(\tan^{-1}\) is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Since \(\tan\frac{\pi}{3} = \sqrt{3}\) and tangent is an odd function, \(\tan^{-1}(-\sqrt{3}) = -\dfrac{\pi}{3}\).
Step 3: Find sec inverse of 2.
The principal range of \(\sec^{-1}\) is \([0, \pi] - \left\{\frac{\pi}{2}\right\}\).
Since \(\sec\frac{\pi}{3} = 2\), we get \(\sec^{-1} 2 = \dfrac{\pi}{3}\).
Step 4: Substitute and evaluate.
\[ \sin\left(-\frac{\pi}{3} - \frac{\pi}{3}\right) = \sin\left(-\frac{2\pi}{3}\right) = -\sin\frac{2\pi}{3} \]
Now \(\sin\frac{2\pi}{3} = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
Final Answer:
The value is \(-\dfrac{\sqrt{3}}{2}\). \[ \boxed{-\frac{\sqrt{3}}{2}} \]
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