Question 1 - Jee advanced Math 2022 P2 Questions with Solutions

 Let \(\alpha\) and \(\beta\) be real numbers such that \(-\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4}\). If \(\sin{(\alpha + \beta)} = \frac{1}{3}\) and \(\cos{(\alpha - \beta)} = \frac{2}{3}\), then the greatest integer less than or equal to 

\((\frac{\sin{\alpha}}{\cos{\beta}} + \frac{\cos{\beta}}{\sin{\alpha}} + \frac{\cos{\alpha}}{\sin{\beta}} + \frac{\sin{\beta}}{\cos{\alpha}})^{2}\)


is ______.


Sol : 


\(\sin{(\alpha + \beta)} = \frac{1}{3}\)

\(\implies \cos^{2}{(\alpha + \beta)} = 1 - \frac{1}{9} = \frac{8}{9}\)


\(\cos{(\alpha - \beta)} = \frac{2}{3}\)

\(\implies \sin^{2}{(\alpha - \beta)} = 1 - \frac{4}{9} = \frac{5}{9}\)


Re-grouping the terms(1 & 3 together and 2 & 4 together) in the given trigonometric expression: 


\( = ((\frac{\sin{\alpha}}{\cos{\beta}} + \frac{\cos{\alpha}}{\sin{\beta}}) + (\frac{\cos{\beta}}{\sin{\alpha}} + \frac{\sin{\beta}}{\cos{\alpha}}))^{2}\)


\( = ((\frac{\sin{\alpha} \sin{\beta} + \cos{\alpha} \cos{\beta}}{\sin{\beta} \cos{\beta}}) + (\frac{\sin{\alpha} \sin{\beta} + \cos{\alpha} \cos{\beta}}{\sin{\alpha} \cos{\alpha}}))^{2}\)


\( = (\frac{\cos{(\alpha - \beta})}{\frac{1}{2}\sin{2\beta}} + \frac{\cos{(\alpha - \beta})}{\frac{1}{2}\sin{2\alpha}})^{2}\)


\( = (2\cos{(\alpha - \beta)} (\frac{\sin{2\alpha} + \sin{2\beta}}{\sin{2\alpha}\sin{2\beta}}) )^{2}\)


\( = (2\cos{(\alpha - \beta)} (\frac{2\sin{(\alpha + \beta)} \cos{(\alpha - \beta)}}{\frac{1}{2}(\cos{2(\alpha - \beta)} - \cos{2(\alpha + \beta)})}))^{2}\)


\( = (2 \cdot 2 \cdot 2 \sin{(\alpha + \beta)}\cos^{2}{(\alpha - \beta)})^{2}(\frac{1}{\cos^{2}{(\alpha - \beta)} - \sin^{2}{(\alpha - \beta)} - \cos^{2}{(\alpha + \beta)} + \sin^{2}{(\alpha + \beta)}})^{2}\)


\( = (2 \cdot 2 \cdot 2 \cdot \frac{1}{3} \cdot \frac{4}{9})^{2}(\frac{1}{\frac{4}{9} - \frac{5}{9} - \frac{8}{9} + \frac{1}{9}})^{2}\)


\( = (\frac{8 \cdot 4}{9 \cdot 3})^{2}(\frac{9}{8})^{2}\)


\( = \frac{16}{9}\)


\( \approx 1.77\)


Therefore, the greatest integer less than or equal to the given expression is 1.

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