Since ![{\bf P}^n](http://latex.mathoverflow.net/png?%7B%5Cbf%20P%7D%5En) is the quotient of ![{\bf A}^{n+1} - 0](http://latex.mathoverflow.net/png?%7B%5Cbf%20A%7D%5E%7Bn%2B1%7D%20%2D%200) by the action of ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm), the tangent bundle of ![{\bf P}^n](http://latex.mathoverflow.net/png?%7B%5Cbf%20P%7D%5En) is the quotient of the tangent bundle of ![{\bf A}^{n+1} - 0](http://latex.mathoverflow.net/png?%7B%5Cbf%20A%7D%5E%7Bn%2B1%7D%20%2D%200) by the action of the tangent bundle of ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm):

![T {\bf P}^n = T ({\bf A}^{n+1} - 0) / T {\bf G}\sb m](http://latex.mathoverflow.net/png?T%20%7B%5Cbf%20P%7D%5En%20%3D%20T%20%28%7B%5Cbf%20A%7D%5E%7Bn%2B1%7D%20%2D%200%29%20%2F%20T%20%7B%5Cbf%20G%7D%5Fm) .

As a group, the tangent bundle of ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm) is the product of ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm) and a 1-dimensional vector space V.  Therefore we can take the quotient of everything on the right side above by ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm).  Note that ![T({\bf A}^{n+1}-0)](http://latex.mathoverflow.net/png?T%20%28%20%7B%5Cbf%20A%7D%5E%7Bn%2B1%7D%20%2D%200%20%29%0A) is the product of ![{\bf A}^{n+1} - 0](http://latex.mathoverflow.net/png?%7B%5Cbf%20A%7D%5E%7Bn%2B1%7D%20%2D%200) with the direct sum of (n+1) copies of the weight one representation of ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm).  Therefore its quotient by ![{\bf G}\sb m](http://latex.mathoverflow.net/png?%7B%5Cbf%20G%7D%5Fm) is ![\mathcal{O}\sb {{\bf P}^n}(1)^{n+1}](http://latex.mathoverflow.net/png?%5Cmathcal%7BO%7D%5F%7B%7B%5Cbf%20P%7D%5En%7D%281%29%5E%7Bn%2B1%7D%0A).  We get

![T {\bf P}^n = \mathcal{O}\sb {{\bf P}^n}(1)^{n+1} / V](http://latex.mathoverflow.net/png?T%20%7B%5Cbf%20P%7D%5En%20%3D%20%5Cmathcal%7BO%7D%5F%7B%7B%5Cbf%20P%7D%5En%7D%281%29%5E%7Bn%2B1%7D%20%2F%20V%0A) .

A (linear) action of a one-dimensional vector space on a vector bundle is the same thing as an exact sequence, so this gives the Euler sequence.