The familiar shapes of atomic orbitals arise from spherical harmonics $Y_{\nabla}^{m}$ or their linear combinations. Given:
$Y_1^1 = (\frac{3}{8 \pi})^{1/2} \sin \theta e^{i\phi}$
$Y_1^{-1} = (\frac{3}{8 \pi})^{1/2} \sin \theta e^{-i\phi}$
Show that the Linear combination $ I(\theta, \phi) = \frac{1}{\sqrt{2i}} (Y_1^1 - Y_1^{-1} ) $ yields a real function.
I found the linear combination of $Y_1^1 - Y_1^{-1} = (e^{i\phi} - e^{-i\phi} )^2 = 2i^2 \sin \phi $
How do I carry on from here?