Simplifying a Nested Trigonometric Expression
Simplify the expression \(\sqrt{1-\cos^2(x)} \cdot \sin(2x)\) completely, and express your result in its simplest algebraic form. Assume \(\sin(x) \geq 0\).
Compound Angle Identities: Simplification and Equation Solving
Consider the following two tasks: (a) Express \(\sin(75^\circ)\) in exact form using the identity \(\sin(A+B) = \sin A \cos B + \cos A \sin B\), with \(A = 45^\circ\) and \(B = 30^\circ\). (b) Solve the equation \(\cos(2\theta) = 0.5\) for \(\theta\) in the interval \([0^\circ, 360^\circ)\).