An exact value for $\sin 15^{\circ}$...

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Without using tables, show for \(\theta=15^\circ\) that \(\sin\theta=\dfrac{\sqrt{6}-\sqrt{2}}{4}\), and find the values of \(\cos\theta\) and \(\tan\theta\).

In the triangle \(ABC\), \(B\) is a right angle and the other angles are such that if a point \(E\) on \(BC\) and between \(B\) and \(C\) is chosen so that \(BE=BA\), then there is a point \(D\) on \(AC\) and between \(A\) and \(C\) such that \(AD=DE=EC\). Find the ratio of \(DC\) to \(AB\).