A ladder rests against a wall …

Let PQ be the ladder such that is top Q is on the wall OQ.
The ladder is pulled away from the wall through a distance a, so Q slides and takes position Q'.
Clearly, {tex}PQ = P'Q'.{/tex}
In {tex}\Delta 's{/tex} {tex}POQ \ and \ P'OQ', {/tex}we have
{tex}\sin \alpha = \frac{{OQ}}{{PQ}},\cos \alpha = \frac{{OP}}{{PQ}},\sin \beta = \frac{{OQ'}}{{P'Q'}},\cos \beta = \frac{{OP'}}{{P'Q'}}{/tex}
{tex} \Rightarrow \sin \alpha = \frac{{b + y}}{{PQ}},\cos \alpha \frac{x}{{PQ}},\sin \beta = \frac{y}{{PQ}},\cos \beta = \frac{{a + x}}{{PQ}}{/tex}
{tex} \Rightarrow \sin \alpha - \sin \beta = \frac{{b + y}}{{PQ}} - \frac{y}{{PQ}}{/tex} and 
{tex}\cos \beta - \cos \alpha = \frac{{a + x}}{{PQ}} - \frac{x}{{PQ}}{/tex}
{tex} \Rightarrow \sin \alpha - \sin \beta = \frac{b}{{PQ}}{/tex} and 
{tex}\cos \beta - \cos \alpha = \frac{a}{{PQ}}{/tex}
{tex} \Rightarrow \frac{a}{b} = \frac{{\cos \alpha - \cos \beta }}{{\sin \beta - \sin \alpha }}{/tex}