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I got a problem with one recast of an expression which pops up by considering the completeness relation of common spherical harmonics:

[tex]\sum_{l=0}^{\infty} \sum_{m=-l}^{+l} Y_{lm}(\theta,\phi)Y^{*}_{lm}(\theta^{\prime},\phi^{\prime}) = \frac{1}{sin(\theta)} \delta(\theta - \theta^{\prime}) \delta(\phi - \phi^{\prime})[/tex]

The question is about the rhs which is sometimes recast like

[tex]\frac{1}{sin(\theta)} \delta(\theta - \theta^{\prime}) \delta(\phi - \phi^{\prime}) = \delta(\cos(\theta) - \cos(\theta^{\prime})) \delta(\phi - \phi^{\prime})[/tex]

After several attempts I just can't explain how to justify that rearrangement. Could you help me please ?