If the tangent at any point of the ellipse $\frac{x^2}{a^3}+\frac{y^2}{b^2}=1$ makes an angle $\alpha$ with the major axis and an angle $\beta$ with the focal radius of the point of contact, then show that the eccentricity of the ellipse is given by $e=\frac{\cos \beta }{\cos \alpha }$

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