Comparing the simplified expressions from both sides:
\[ \sqrt{3} + 3 \stackrel{?}{=} -\sqrt{3} \]
The expressions are not equal, so there seems to be an error in the provided equation. Please double-check the equation or clarify if there was a mistake in transcription.
First, find the value of \(\cot 30^\circ\), which is \(\sqrt{3}\):
\[ (\sqrt{3} + 1) \cdot (3 - \sqrt{3}) \]
Now, expand the expression:
\[ 3\sqrt{3} - 3 + \sqrt{3} - 1 \]
Combine like terms:
\[ 4\sqrt{3} - 4 \]
**Right Side:**
\[ \tan 360^\circ - 2 \sin 60^\circ \]
Since \(\tan 360^\circ\) is equivalent to \(\tan 0^\circ\), which is 0, and \(\sin 60^\circ\) is \(\frac{\sqrt{3}}{2}\):
\[ 0 - 2 \cdot \frac{\sqrt{3}}{2} \]
Combine like terms:
\[ -\sqrt{3} \]
Now, compare the left side and the right side:
\[ 4\sqrt{3} - 4 \neq -\sqrt{3} \]
The equation is not satisfied, and thus, the given statement is incorrect. There may be a mistake in the provided equation. Please double-check the equation for accuracy.
Answers & Comments
Answer:
Let's break down the expression step by step:
Given expression: \( (\sqrt{3} + 1)(3 - \cot 30^\circ) \)
First, simplify the expression within the parentheses:
\[ 3 - \cot 30^\circ = 3 - \frac{\cos 30^\circ}{\sin 30^\circ} \]
\[ = 3 - \frac{\sqrt{3}/2}{1/2} = 3 - \sqrt{3} \]
Now, substitute this back into the original expression:
\[ (\sqrt{3} + 1)(3 - \sqrt{3}) \]
Expand the expression:
\[ 3\sqrt{3} + \sqrt{3} + 3 - 3\sqrt{3} \]
Combine like terms:
\[ \sqrt{3} + 3 \]
Now, simplify the right side of the given expression:
\[ \tan 360^\circ - 2\sin 60^\circ \]
Since \(\tan 360^\circ = \tan 0^\circ = 0\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\):
\[ 0 - 2\left(\frac{\sqrt{3}}{2}\right) = -\sqrt{3} \]
Comparing the simplified expressions from both sides:
\[ \sqrt{3} + 3 \stackrel{?}{=} -\sqrt{3} \]
The expressions are not equal, so there seems to be an error in the provided equation. Please double-check the equation or clarify if there was a mistake in transcription.
Step-by-step explanation:
To prove the given equation:
\[ (\sqrt{3} + 1) \cdot (3 - \cot 30^\circ) = \tan 360^\circ - 2 \sin 60^\circ \]
Let's simplify each side step by step:
**Left Side:**
\[ (\sqrt{3} + 1) \cdot (3 - \cot 30^\circ) \]
First, find the value of \(\cot 30^\circ\), which is \(\sqrt{3}\):
\[ (\sqrt{3} + 1) \cdot (3 - \sqrt{3}) \]
Now, expand the expression:
\[ 3\sqrt{3} - 3 + \sqrt{3} - 1 \]
Combine like terms:
\[ 4\sqrt{3} - 4 \]
**Right Side:**
\[ \tan 360^\circ - 2 \sin 60^\circ \]
Since \(\tan 360^\circ\) is equivalent to \(\tan 0^\circ\), which is 0, and \(\sin 60^\circ\) is \(\frac{\sqrt{3}}{2}\):
\[ 0 - 2 \cdot \frac{\sqrt{3}}{2} \]
Combine like terms:
\[ -\sqrt{3} \]
Now, compare the left side and the right side:
\[ 4\sqrt{3} - 4 \neq -\sqrt{3} \]
The equation is not satisfied, and thus, the given statement is incorrect. There may be a mistake in the provided equation. Please double-check the equation for accuracy.