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Let's solve the given expression step by step:

The expression is:

\[ \cos 45^\circ \cdot \sin 45^\circ + \tan 60^\circ \cdot \cot 60^\circ \]

We start by finding the values of each trigonometric function:

1. \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\)
2. \(\tan 60^\circ = \sqrt{3}\)
3. \(\cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}}\)

Now, substitute these values into the expression:

\[ \cos 45^\circ \cdot \sin 45^\circ = \left(\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{\sqrt{2}}{2}\right) = \frac{2}{4} = \frac{1}{2} \]

\[ \tan 60^\circ \cdot \cot 60^\circ = \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 \]

Now, sum the results:

\[ \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]

So, the value of the given expression is \(\frac{3}{2}\).

Therefore, the correct answer is B) \(\frac{3}{2}\).

Do you want any details or have any questions? 

Here are 5 more questions you might find interesting:

1. How do you calculate \(\sin 30^\circ \cdot \cos 30^\circ\)?
2. What is the value of \(\tan 45^\circ + \cot 45^\circ\)?
3. Can you simplify \(\sin^2 x + \cos^2 x\) for any angle \(x\)?
4. How do you convert degrees to radians?
5. What is the exact value of \(\cos 60^\circ\)?

Tip: Always remember the key trigonometric identities, as they can simplify solving many problems.

Learn how to solve the trigonometric expression cos 45° * sin 45° + tan 60° * cot 60° step-by-step. This problem involves trigonometric identities and calculations of specific trigonometric functions like cosine, sine, tangent, and cotangent. Suitable for Grade 9 students.

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