\(x\sin\alpha + y\sin\alpha = P_1\)
\(x\cos\alpha + y\sin\alpha = P_2\)
Let Angle between both line is \(\theta\)
\(\therefore cos\theta = \frac{\vec {b_1}.\vec{b_2}}{|\vec{b_1}||\vec {b_2}|}\)
\(= \frac{\sin\alpha .\cos\alpha + \sin^2x}{\sqrt 2\sin\alpha \times 1}\)
\(= \frac{\sin\alpha (\sin\alpha + \cos \alpha)}{\sqrt 2 \sin \alpha}\)
\(= \frac{\sin\alpha + \cos\alpha}{\sqrt 2}\)
\(= \cos \alpha \cos45° + \sin \alpha \sin 45° \)
\(=\cos (\alpha - 45°)\)
\(\therefore \theta = \alpha - 45°\)
