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Trigonometric substitution [migrated]

Been out of touch with trigonometry for some time now. Need help proving this expression. Sin2x/2 = 1/2(1-Cosx) Any help will be appreciated. Thanks.
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Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
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Existence of nonergodic polygonal billiard

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one. A standard conjecture is that a ...
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Maximum chi-square distance between norm vectors [closed]

What is the maximum possible chi-square distance between two normalized vectors? The representation of chi-square distance is below. $d(x,y) = \sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$
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Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by: $$\log(\sum_{i=0}^n |z_i|^2)).$$ What is the analogous formula for a Kaehler ...
Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...