Let's break down and simplify the given expression: 

\[
\cos 360^\circ - (\tan 45^\circ \times \cot 90^\circ) + \log (\sin 60^\circ)
\]

### Step 1: Evaluate individual trigonometric values:
1. \(\cos 360^\circ = 1\)
2. \(\tan 45^\circ = 1\)
3. \(\cot 90^\circ = \frac{1}{\tan 90^\circ} = 0\) 
   (since \(\tan 90^\circ\) is undefined, \(\cot 90^\circ\) is 0)
4. \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)

### Step 2: Simplify the expression:
\[
\cos 360^\circ - (\tan 45^\circ \times \cot 90^\circ) + \log (\sin 60^\circ)
\]
\[
= 1 - (1 \times 0) + \log \left(\frac{\sqrt{3}}{2}\right)
\]
\[
= 1 - 0 + \log \left(\frac{\sqrt{3}}{2}\right)
\]

### Step 3: Express the logarithmic part:
Using logarithmic properties:
\[
\log \left(\frac{\sqrt{3}}{2}\right) = \log \sqrt{3} - \log 2 = \frac{1}{2} \log 3 - \log 2
\]

### Final Answer:
\[
1 + \left(\frac{1}{2} \log 3 - \log 2\right)
\]
This is the simplified form of the given expression.

Do you want more details or have any questions about this solution?

Here are five related questions you might find interesting:
1. What is the value of \(\cos 90^\circ\) and why?
2. How do you find the cotangent of an angle, and why is \(\cot 90^\circ\) zero?
3. Can you explain how logarithmic properties help simplify complex expressions?
4. Why is the tangent of 45 degrees equal to 1?
5. How does the unit circle relate to calculating trigonometric functions like \(\cos 360^\circ\)?

**Tip:** Remember, \(\cot \theta\) is the reciprocal of \(\tan \theta\), and this fact is useful when simplifying trigonometric expressions.

Explore how to simplify the expression Cos 360 - (Tan 45 * Cot 90) + log(Sin 60) with step-by-step explanations. This problem covers essential trigonometric values, relationships, and logarithmic properties, making it suitable for students in Grades 9-11.

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