4) Given that tan(pi )/(3)=sqrt (3) use an equivalent trigonometric expr ession (related or corelated) to verify that cot(pi )/(6)=sqrt (3)

## Step 1: Understand the Relationship Between Tangent and Cotangent<br />### The cotangent function is the reciprocal of the tangent function. Therefore, if \(\tan \theta = x\), then \(\cot \theta = \frac{1}{x}\). This relationship will help us verify the given expression.<br />## Step 2: Use the Given Information<br />### We are given that \(\tan \frac{\pi}{3} = \sqrt{3}\). Since \(\cot \theta = \frac{1}{\tan \theta}\), we can find \(\cot \frac{\pi}{3}\) as \(\frac{1}{\sqrt{3}}\).<br />## Step 3: Relate Angles Using Complementary Angles<br />### The angles \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\) are complementary because their sum is \(\frac{\pi}{2}\). For complementary angles, \(\tan \theta = \cot (\frac{\pi}{2} - \theta)\).<br />## Step 4: Verify \(\cot \frac{\pi}{6}\)<br />### Since \(\tan \frac{\pi}{3} = \sqrt{3}\), it follows that \(\cot \frac{\pi}{6} = \tan \frac{\pi}{3} = \sqrt{3}\).