The value of α32cosec2⁡(12tan−1⁡αβ)+β32sec2⁡(12tan−1⁡(βα)) is equal to

α32cosec2⁡(12tan−1⁡αβ)+β32sec2⁡(12tan−1⁡βα)=α311−cos⁡(tan−1⁡(αβ))+β311+cos⁡(tan−1⁡βα)

The value of α32cosec2⁡(12tan−1⁡αβ)+β32sec2⁡(12tan−1⁡(βα)) is equal to

α32cosec2⁡(12tan−1⁡αβ)+β32sec2⁡(12tan−1⁡βα)=α311−cos⁡(tan−1⁡(αβ))+β311+cos⁡(tan−1⁡βα)

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The value of $\frac{α^{3}}{2} {cosec}^{2} (\frac{1}{2} \tan^{- 1} \frac{α}{β}) + \frac{β^{3}}{2} \sec^{2} (\frac{1}{2} \tan^{- 1} (\frac{β}{α}))$ is equal to

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$(α - β) (α^{2} - β^{2})$

$(α - β) (α^{2} + β^{2})$

$(α + β) (α^{2} - β^{2})$

$(α + β) (α^{2} + β^{2})$

answer is C.

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Detailed Solution

$\frac{α^{3}}{2} {cosec}^{2} (\frac{1}{2} \tan^{- 1} \frac{α}{β}) + \frac{β^{3}}{2} \sec^{2} (\frac{1}{2} \tan^{- 1} \frac{β}{α})$

$= α^{3} \frac{1}{1 - \cos (\tan^{- 1} (\frac{α}{β}))} + β^{3} \frac{1}{1 + \cos (\tan^{- 1} \frac{β}{α})}$

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The value of α32cosec2⁡(12tan−1⁡αβ)+β32sec2⁡(12tan−1⁡(βα)) is equal to

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The value of $\frac{α^{3}}{2} {cosec}^{2} (\frac{1}{2} \tan^{- 1} \frac{α}{β}) + \frac{β^{3}}{2} \sec^{2} (\frac{1}{2} \tan^{- 1} (\frac{β}{α}))$ is equal to