Answer
$1.15, 3.57$
Work Step by Step
Step 1. Use the Half-Angle Formulas $sin\theta=\sqrt {\frac{1-cos2\theta}{2}}$ and $cos\theta=\sqrt {\frac{1+cos2\theta}{2}}$, we transform the equation as $\sqrt {\frac{1-cos2\theta}{2}}-\sqrt {\frac{1+cos2\theta}{2}}=\frac{1}{2}$
Step 2. Take the square of both sides of the equation to get
$\frac{1-cos2\theta}{2}+\frac{1+cos2\theta}{2}-\sqrt {1-cos^22\theta}=\frac{1}{4}$ which gives $\sqrt {1-cos^22\theta}=\frac{3}{4}$
Step 3. Again take the square of both sides to get $1-cos^22\theta=\frac{9}{16}$ which leads to
$cos^22\theta=\frac{7}{16}$ thus $cos2\theta=\pm\frac{\sqrt 7}{4}$
Step 4. Use a calculator, the solutions to the above equation within $[0,4\pi)$ can be found as
$2\theta=0.848, 2.294, 7.131, 8,577$ thus we get $\theta=0.42, 1.15, 3.57, 4.29$ plug-in these numbers
into the original equation, we get the final answers as $\theta= 1.15, 3.57$
