Ricci tensor for Schwarzschild metric

Good morning. I have calculated the components of the Ricci tensor for the Schwarzschild metric and they are all 0, except the last component R33. I have calculated it over and over again, and I keep getting -2(sinΘ)^2. Does anyone know how it vanishes? I accounted for every index, and my Christoffel symbols are correct, but it's not vanishing. Also, for the ø component of the Schwarzschild metric, is it (r^2)(sinΘ)^2 or (r^2)(cosΘ)^2 ? Leonard Susskind uses cosine, but everywhere else I read it's sine. Which is it?



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