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I have a short dirac delta type question, which involves a dirac delta composite function. Can anyone see how the following is equivalent (given in bra-ket notation): $$| \theta_0, \phi_0 \rangle \langle \theta_0, \phi_0| = \int \sin \theta d \theta d \phi \delta(\theta -\theta_0) \delta[\sin \theta( \phi - \phi_0)]| \theta, \phi \rangle \langle \theta, \phi |$$

Thanks.

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Note that $$\int_{-\infty}^\infty dx \delta(ax)f(x)=\frac{1}{a}f(0)$$ Then, integrating over $\phi$ $$\int \sin \theta d \theta \delta(\theta -\theta_0) \int d \phi \delta[\sin \theta( \phi - \phi_0)]| \theta, \phi \rangle \langle \theta, \phi |=\int \sin \theta d \theta \frac{1}{\sin\theta}\delta(\theta -\theta_0) | \theta, \phi_0 \rangle \langle \theta, \phi_0 |$$ Now integrating over theta, yields $$\int d \theta \delta(\theta -\theta_0) | \theta, \phi_0 \rangle \langle \theta, \phi_0 |= | \theta_0, \phi_0 \rangle \langle \theta_0, \phi_0 |$$

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