If tan⁡ and then tan⁡ (where and = both are acute angles)

JEE MainMathematicsEasyDefinite Integration

If tan⁡ $(\alpha + \beta) = \frac{1}{2}\$ and $tan(\alpha = \beta) = \frac{1}{3}$ then tan⁡ $2\alpha\ is\$ (where $\alpha + \beta$ and $\alpha$
= $\beta\$ both are acute angles)

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tan2\alpha = tan\lbrack(\alpha + \beta) + (\alpha - \beta)\rbrack

= \frac{\tan(\alpha + \beta) + \tan(\alpha - \beta)}{1 - \tan(\alpha - \beta)\tan(\alpha + \beta)}

= \frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2}\ \ \frac{1}{3}\ } = 1

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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