If `cosA=mcosB`, then

Options

A. `cot""(A+B)/(2)=(m+1)/(m-1)tan""(B-A)/(2)`
B. `tan""(A+B)/(2)=(m+1)/(m-1)cot""(B-A)/(2)`
C. `cot""(A+B)/(2)=(m+1)/(m-1)tan""(A-B)/(2)`
D. `cot""(A+B)/(2)=(m-1)/(m+1)tan""(A-B)/(2)`

Answer

To solve the equation given \( \cos A = m \cos B \), we will derive the relationships step by step.

### Step 1: Rearranging the equation
We start with the equation:
\[ \cos A = m \cos B \]
Dividing both sides by \( \cos B \) (assuming \( \cos B \neq 0 \)):
\[ \frac{\cos A}{\cos B} = m \]

**Hint:** Remember that dividing by a trigonometric function is valid only if that function is not zero.

### Step 2: Applying the Cosine Addition Formula
Using the cosine addition formula:
\[ \cos A = \cos(B + (A - B)) = \cos B \cos(A - B) - \sin B \sin(A - B) \]
We can express \( \cos A \) in terms of \( \cos B \) and \( \sin B \).

**Hint:** Familiarize yourself with the cosine addition and subtraction formulas to manipulate cosine expressions.

### Step 3: Using the Compound Angle Formula
Now, we can apply the compound angle formula for cosine:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
and
\[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
This gives us two equations to work with.

**Hint:** The compound angle formulas help simplify expressions involving sums and differences of angles.

### Step 4: Setting up the equation
From the rearranged equation, we can express it as:
\[ \frac{\cos A + \cos B}{\cos A - \cos B} = \frac{m + 1}{m - 1} \]

**Hint:** This step involves recognizing how to structure the equation to isolate the terms involving \( m \).

### Step 5: Substituting the formulas
Substituting the cosine formulas into our equation:
\[ \frac{2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)}{-2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)} = \frac{m + 1}{m - 1} \]
This simplifies to:
\[ \frac{\cos\left(\frac{A + B}{2}\right)}{\sin\left(\frac{A + B}{2}\right)} = \frac{m + 1}{m - 1} \]

**Hint:** Recognize that the left side can be expressed as cotangent.

### Step 6: Final expression
Thus, we have:
\[ \cot\left(\frac{A + B}{2}\right) = \frac{m + 1}{m - 1} \tan\left(\frac{B - A}{2}\right) \]

**Hint:** This is the final form of the relationship derived from the original equation.

### Conclusion
The derived relationship shows how \( \cot\left(\frac{A + B}{2}\right) \) relates to \( m \) and the tangent of the difference of angles. This can be used to analyze various properties of angles \( A \) and \( B \).