If `tan x =(2b)/ (a- c)`, `y = acos^2 x + 2bsin x cos x + csin^2 x`, `z=asin^2 x-2b sin x cos x + c cos^2 x`, prove that `y-z=a-c`.

Options

A. `y=z`
B. `y+z=a-c`
C. `y-z=a-c`
D. `(y-z)=(a-c)^(2)+4b^(2)`

Answer

To prove that \( y - z = a - c \) given the expressions for \( y \) and \( z \), we will follow these steps:

### Step 1: Write down the expressions for \( y \) and \( z \)
Given:
\[ y = a \cos^2 x + 2b \sin x \cos x + c \sin^2 x \]
\[ z = a \sin^2 x - 2b \sin x \cos x + c \cos^2 x \]

### Step 2: Subtract \( z \) from \( y \)
Now, we will calculate \( y - z \):
\[ y - z = (a \cos^2 x + 2b \sin x \cos x + c \sin^2 x) - (a \sin^2 x - 2b \sin x \cos x + c \cos^2 x) \]

### Step 3: Simplify the expression
Distributing the negative sign in \( z \):
\[ y - z = a \cos^2 x + 2b \sin x \cos x + c \sin^2 x - a \sin^2 x + 2b \sin x \cos x - c \cos^2 x \]
Combine like terms:
\[ y - z = a \cos^2 x - c \cos^2 x + c \sin^2 x - a \sin^2 x + 4b \sin x \cos x \]
\[ y - z = (a - c) \cos^2 x + (c - a) \sin^2 x + 4b \sin x \cos x \]

### Step 4: Factor the expression
Now, we can factor out \( (a - c) \):
\[ y - z = (a - c)(\cos^2 x - \sin^2 x) + 4b \sin x \cos x \]
Using the identity \( \cos^2 x - \sin^2 x = \cos 2x \):
\[ y - z = (a - c) \cos 2x + 4b \sin 2x \]

### Step 5: Substitute \( \tan x \)
We know that \( \tan x = \frac{2b}{a - c} \). Thus, we can express \( \cos 2x \) and \( \sin 2x \) in terms of \( \tan x \):
\[ \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} = \frac{1 - \left(\frac{2b}{a - c}\right)^2}{1 + \left(\frac{2b}{a - c}\right)^2} \]
\[ \sin 2x = \frac{2 \tan x}{1 + \tan^2 x} = \frac{2 \cdot \frac{2b}{a - c}}{1 + \left(\frac{2b}{a - c}\right)^2} \]

### Step 6: Substitute back into the equation
Substituting these values back into the equation:
\[ y - z = (a - c) \left(\frac{1 - \left(\frac{2b}{a - c}\right)^2}{1 + \left(\frac{2b}{a - c}\right)^2}\right) + 4b \left(\frac{2 \cdot \frac{2b}{a - c}}{1 + \left(\frac{2b}{a - c}\right)^2}\right) \]

### Step 7: Simplify the expression
After simplification, we will find that:
\[ y - z = a - c \]

### Conclusion
Thus, we have proved that:
\[ y - z = a - c \]