Question:
The value of
\[
\sin^{-1}\left[\cos\left(\sin^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right) + \cos^{-1}\left(\frac{\sqrt{12}}{4}\right) + \sec^{-1}\left(\sqrt{2}\right)\right)\right]
\]
is
The value of
\[
\sin^{-1}\left[\cos\left(\sin^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right) + \cos^{-1}\left(\frac{\sqrt{12}}{4}\right) + \sec^{-1}\left(\sqrt{2}\right)\right)\right]
\]
is
Show Hint
The value of $\sin\sqrt2-\sqrt3/4 + \cos\sqrt12/4 + \sec\sqrt2)]$ is
Updated On: Apr 15, 2026
- 0
- $\pi/4$
- $\pi/6$
- $\pi/2$
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Evaluate the standard inverse trigonometric values.
Step 2: Analysis
$\sec^{-1}\sqrt{2} = 45^\circ$, $\cos^{-1}\frac{\sqrt{12}}{4} = \cos^{-1}\frac{\sqrt{3}}{2} = 30^\circ$. $\sqrt{\frac{2-\sqrt{3}}{4}} = \frac{\sqrt{3}-1}{2\sqrt{2}} = \sin 15^\circ$.
Step 3: Evaluation
The expression is $\sin^{-1}[\cos(15^\circ + 30^\circ + 45^\circ)] = \sin^{-1}[\cos 90^\circ]$.
Step 4: Conclusion
$\sin^{-1}(0) = 0$.
Final Answer: (a)
Evaluate the standard inverse trigonometric values.
Step 2: Analysis
$\sec^{-1}\sqrt{2} = 45^\circ$, $\cos^{-1}\frac{\sqrt{12}}{4} = \cos^{-1}\frac{\sqrt{3}}{2} = 30^\circ$. $\sqrt{\frac{2-\sqrt{3}}{4}} = \frac{\sqrt{3}-1}{2\sqrt{2}} = \sin 15^\circ$.
Step 3: Evaluation
The expression is $\sin^{-1}[\cos(15^\circ + 30^\circ + 45^\circ)] = \sin^{-1}[\cos 90^\circ]$.
Step 4: Conclusion
$\sin^{-1}(0) = 0$.
Final Answer: (a)
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