The angle between the lines `(x^(2)+y^(2))sin^(2)alpha=(x cos beta-y sin beta)^(2)` is

Answer

To find the angle between the lines given by the equation \((x^2 + y^2) \sin^2 \alpha = (x \cos \beta - y \sin \beta)^2\), we can follow these steps:

### Step 1: Rewrite the given equation
Start with the original equation:
\[ (x^2 + y^2) \sin^2 \alpha = (x \cos \beta - y \sin \beta)^2 \]

### Step 2: Expand the right-hand side
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\), we expand the right-hand side:
\[ (x \cos \beta - y \sin \beta)^2 = x^2 \cos^2 \beta - 2xy \cos \beta \sin \beta + y^2 \sin^2 \beta \]
Thus, we have:
\[ (x^2 + y^2) \sin^2 \alpha = x^2 \cos^2 \beta - 2xy \cos \beta \sin \beta + y^2 \sin^2 \beta \]

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ x^2 \sin^2 \alpha + y^2 \sin^2 \alpha - x^2 \cos^2 \beta - y^2 \sin^2 \beta + 2xy \cos \beta \sin \beta = 0 \]
This can be simplified to:
\[ x^2 (\sin^2 \alpha - \cos^2 \beta) + y^2 (\sin^2 \alpha - \sin^2 \beta) + 2xy \cos \beta \sin \beta = 0 \]

### Step 4: Identify coefficients
From the standard form of a pair of lines \(Ax^2 + 2Hxy + By^2 = 0\), we can identify:
- \(A = \sin^2 \alpha - \cos^2 \beta\)
- \(B = \sin^2 \alpha - \sin^2 \beta\)
- \(H = \cos \beta \sin \beta\)

### Step 5: Use the formula for the angle between the lines
The formula for the angle \(\theta\) between the two lines is given by:
\[ \tan \theta = \frac{|2H|}{A + B} \]

### Step 6: Substitute the coefficients into the formula
Substituting the values of \(A\), \(B\), and \(H\):
\[ \tan \theta = \frac{|2 \cos \beta \sin \beta|}{(\sin^2 \alpha - \cos^2 \beta) + (\sin^2 \alpha - \sin^2 \beta)} \]

### Step 7: Simplify the expression
The denominator simplifies to:
\[ 2 \sin^2 \alpha - (\cos^2 \beta + \sin^2 \beta) = 2 \sin^2 \alpha - 1 \]
Thus, we have:
\[ \tan \theta = \frac{2 |\cos \beta \sin \beta|}{2 \sin^2 \alpha - 1} \]

### Step 8: Use trigonometric identities
Using the identities \(\sin 2\beta = 2 \sin \beta \cos \beta\) and \(\cos 2\alpha = 1 - 2 \sin^2 \alpha\), we can rewrite:
\[ \tan \theta = \frac{|\sin 2\beta|}{-\cos 2\alpha} \]

### Step 9: Final result
This leads us to:
\[ \tan \theta = \tan 2\alpha \]
Thus, the angle \(\theta\) between the lines is:
\[ \theta = 2\alpha \]

### Final Answer
The angle between the lines is \(2\alpha\).

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