Prove-that-x-3-2sin-2-1-2-arctan-x-y-y-3-2cos-2-1-2-arctan-y-x-x-y-x-2-y-2-

Question Number 195157 by Erico last updated on 25/Jul/23

Prove that

\frac{x^{3}}{2 {s i n}^{2} (\frac{1}{2} a r c t a n \frac{x}{y})} + \frac{y^{3}}{2 {c o s}^{2} (\frac{1}{2} a r c t a n \frac{y}{x})} = (x + y) (x^{2} + y^{2})

Answered by Frix last updated on 25/Jul/23

\sin^{2} \tan^{- 1} α = \frac{\sqrt{α^{2} + 1} - 1}{2 \sqrt{α^{2} + 1}} = \frac{\sqrt{x^{2} + y^{2}} - y}{2 \sqrt{x^{2} + y^{2}}}

\cos^{2} \tan^{- 1} \frac{1}{α} = \frac{\sqrt{α^{2} + 1} + α}{2 \sqrt{α^{2} + 1}} = \frac{\sqrt{x^{2} + y^{2}} + x}{2 \sqrt{x^{2} + y^{2}}}

\frac{x^{3}}{2 \frac{\sqrt{x^{2} + y^{2}} - y}{2 \sqrt{x^{2} + y^{2}}}} + \frac{y^{3}}{2 \frac{\sqrt{x^{2} + y^{2}} + x}{2 \sqrt{x^{2} + y^{2}}}} =

= \sqrt{x^{2} + y^{2}} (\frac{x^{3}}{\sqrt{x^{2} + y^{2}} - y} + \frac{y^{3}}{\sqrt{x^{2} + y^{2}} + x}) =

= \sqrt{x^{2} + y^{2}} (x (\sqrt{x^{2} + y^{2}} + y) + y (\sqrt{x^{2} + y^{2}} - x)) =

= \sqrt{x^{2} + y^{2}} (x + y) \sqrt{x^{2} + y^{2}} =

= (x + y) (x^{2} + y^{2})

Facebook Tweet Pin