If ` ( cos x - cos alpha)/(cos x - cos beta) = ( sin^2 alpha cos beta)/(sin^2 beta cos alpha)` then cos x =

Options

A. (a) `cos x=(cos alpha+cos beta)/(1-cos alpha cos beta)`
B. (b) `cos x=(cos alpha-cos beta)/(1-cos alpha cos beta)`
C. (c) `cos x=(cos alpha+cos beta)/(1+cos alpha cos beta)`
D. (d) `cos x=(cos alpha-cos beta)/(1+cos alpha cos beta)`

Explanation

To solve the equation

\[ \frac{\cos x - \cos \alpha}{\cos x - \cos \beta} = \frac{\sin^2 \alpha \cos \beta}{\sin^2 \beta \cos \alpha}, \]

we will follow these steps:

### Step 1: Cross Multiply
We start by cross-multiplying the equation to eliminate the fraction:

\[ (\cos x - \cos \alpha) \sin^2 \beta \cos \alpha = (\cos x - \cos \beta) \sin^2 \alpha \cos \beta. \]

### Step 2: Expand Both Sides
Now, we will expand both sides of the equation:

\[ \cos x \sin^2 \beta \cos \alpha - \cos \alpha \sin^2 \beta \cos \alpha = \cos x \sin^2 \alpha \cos \beta - \cos \beta \sin^2 \alpha \cos \beta. \]

### Step 3: Rearrange Terms
Next, we will rearrange the equation to group all terms involving \(\cos x\) on one side:

\[ \cos x \sin^2 \beta \cos \alpha - \cos x \sin^2 \alpha \cos \beta = \cos \alpha \sin^2 \beta \cos \alpha - \cos \beta \sin^2 \alpha \cos \beta. \]

### Step 4: Factor Out \(\cos x\)
We can factor \(\cos x\) out from the left-hand side:

\[ \cos x (\sin^2 \beta \cos \alpha - \sin^2 \alpha \cos \beta) = \cos \alpha \sin^2 \beta \cos \alpha - \cos \beta \sin^2 \alpha \cos \beta. \]

### Step 5: Solve for \(\cos x\)
Now, we can solve for \(\cos x\):

\[ \cos x = \frac{\cos \alpha \sin^2 \beta \cos \alpha - \cos \beta \sin^2 \alpha \cos \beta}{\sin^2 \beta \cos \alpha - \sin^2 \alpha \cos \beta}. \]

### Step 6: Substitute \(\sin^2\) in Terms of \(\cos^2\)
Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can substitute for \(\sin^2 \alpha\) and \(\sin^2 \beta\):

\[ \sin^2 \alpha = 1 - \cos^2 \alpha, \quad \sin^2 \beta = 1 - \cos^2 \beta. \]

### Step 7: Substitute and Simplify
Substituting these into our equation gives:

\[ \cos x = \frac{\cos \alpha (1 - \cos^2 \beta) \cos \alpha - \cos \beta (1 - \cos^2 \alpha) \cos \beta}{(1 - \cos^2 \beta) \cos \alpha - (1 - \cos^2 \alpha) \cos \beta}. \]

### Step 8: Simplify Further
Now, we can simplify both the numerator and the denominator.

### Final Result
After simplification, we find:

\[ \cos x = \frac{\cos \alpha + \cos \beta}{1 + \cos \alpha \cos \beta}. \]

### Conclusion
Thus, the final answer is:

\[ \cos x = \frac{\cos \alpha + \cos \beta}{1 + \cos \alpha \cos \beta}. \]