Under rotation of axes through `theta` , `x cosalpha + ysinalpha=P` changes to `Xcos beta + Y sin beta=P` then . (a) `cos beta = cos (alpha - theta)` (b) `cos alpha= cos( beta - theta)` (c) `sin beta = sin (alpha - theta)` (d) `sin alpha = sin ( beta - theta)`

Options

A. `cos beta = cos (alpha - theta)`
B. `cos alpha = cos (beta - theta)`
C. `sin beta = sin (alpha - theta)`
D. `sin alpha = sin (beta - theta)`

Explanation

To solve the problem, we need to analyze the transformation of the equation under the rotation of axes. The given equation is:

\[ x \cos \alpha + y \sin \alpha = P \]

After rotation through an angle \( \theta \), the coordinates change to \( X \) and \( Y \), leading to the new equation:

\[ X \cos \beta + Y \sin \beta = P \]

We need to find the relationships between \( \alpha \), \( \beta \), and \( \theta \).

### Step-by-Step Solution:

1. **Substitute the transformations for \( x \) and \( y \)**:
The transformations for the coordinates under rotation are:
\[ x = X \cos \theta - Y \sin \theta \]
\[ y = X \sin \theta + Y \cos \theta \]

2. **Substitute these into the original equation**:
Replace \( x \) and \( y \) in the equation \( x \cos \alpha + y \sin \alpha = P \):
\[ (X \cos \theta - Y \sin \theta) \cos \alpha + (X \sin \theta + Y \cos \theta) \sin \alpha = P \]

3. **Expand the equation**:
Expanding the left-hand side gives:
\[ X \cos \theta \cos \alpha - Y \sin \theta \cos \alpha + X \sin \theta \sin \alpha + Y \cos \theta \sin \alpha = P \]

4. **Rearranging terms**:
Group the terms involving \( X \) and \( Y \):
\[ X (\cos \theta \cos \alpha + \sin \theta \sin \alpha) + Y (\cos \theta \sin \alpha - \sin \theta \cos \alpha) = P \]

5. **Use trigonometric identities**:
Recognize the trigonometric identities:
\[ \cos(\alpha - \theta) = \cos \theta \cos \alpha + \sin \theta \sin \alpha \]
\[ \sin(\alpha - \theta) = \sin \theta \cos \alpha - \cos \theta \sin \alpha \]

Thus, we can rewrite the equation as:
\[ X \cos(\alpha - \theta) + Y \sin(\alpha - \theta) = P \]

6. **Compare with the new equation**:
Now, we compare this with the new equation \( X \cos \beta + Y \sin \beta = P \). From this comparison, we can conclude:
\[ \cos \beta = \cos(\alpha - \theta) \]
\[ \sin \beta = \sin(\alpha - \theta) \]

### Conclusion:
From the derived relationships, we can conclude:
- \( \cos \beta = \cos(\alpha - \theta) \)
- \( \sin \beta = \sin(\alpha - \theta) \)

Thus, the correct options are:
(a) \( \cos \beta = \cos(\alpha - \theta) \) and (c) \( \sin \beta = \sin(\alpha - \theta) \).