The numerical value of `tan 225^circ *cot 405^circ + tan 765^circ* cot 675^circ` =______.

Options

A. 0
B. 1
C. -1
D. 2

Answer

To solve the expression \( \tan 225^\circ \cdot \cot 405^\circ + \tan 765^\circ \cdot \cot 675^\circ \), we will evaluate each term step by step.

### Step 1: Evaluate \( \tan 225^\circ \)

We know that:
\[ \tan 225^\circ = \tan(180^\circ + 45^\circ) = \tan 45^\circ = -1 \]

### Step 2: Evaluate \( \cot 405^\circ \)

Next, we simplify \( \cot 405^\circ \):
\[ \cot 405^\circ = \cot(360^\circ + 45^\circ) = \cot 45^\circ = 1 \]

### Step 3: Calculate \( \tan 225^\circ \cdot \cot 405^\circ \)

Now, we can multiply the results from Step 1 and Step 2:
\[ \tan 225^\circ \cdot \cot 405^\circ = (-1) \cdot 1 = -1 \]

### Step 4: Evaluate \( \tan 765^\circ \)

Now we evaluate \( \tan 765^\circ \):
\[ \tan 765^\circ = \tan(720^\circ + 45^\circ) = \tan 45^\circ = 1 \]

### Step 5: Evaluate \( \cot 675^\circ \)

Next, we simplify \( \cot 675^\circ \):
\[ \cot 675^\circ = \cot(630^\circ + 45^\circ) = \cot 45^\circ = 1 \]

### Step 6: Calculate \( \tan 765^\circ \cdot \cot 675^\circ \)

Now, we multiply the results from Step 4 and Step 5:
\[ \tan 765^\circ \cdot \cot 675^\circ = 1 \cdot 1 = 1 \]

### Step 7: Combine the results

Finally, we combine the two results:
\[ \tan 225^\circ \cdot \cot 405^\circ + \tan 765^\circ \cdot \cot 675^\circ = -1 + 1 = 0 \]

### Final Answer

Thus, the numerical value of the expression is:
\[ \boxed{0} \]