Evaluate: cot-1{2cos(sin-1(√3/2))}
Solution:
Step 1: Evaluate sin⁻¹(√3/2)
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]
Step 2: Substitute
\[ 2\cos\left(\frac{\pi}{3}\right) \]
\[ = 2 \times \frac{1}{2} = 1 \]
Step 3: Apply cot⁻¹
\[ \cot^{-1}(1) \]
We know:
\[ \cot\left(\frac{\pi}{4}\right) = 1 \]
Principal value range of cot⁻¹(x):
\[ (0, \pi) \]
So,
\[ \cot^{-1}(1) = \frac{\pi}{4} \]
Final Answer:
Value = \[ \frac{\pi}{4} \]