Question

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

Answer 1

Explanation:

We have,

`\begin{gathered} \bullet \: \bold{ - (\pi)/(4) < \beta < 0 < \alpha < (\pi)/(4) } \\ \\ \implies \: - (\pi)/(4) < \alpha + \beta < (\pi)/(4) \end{gathered}`

`\begin{gathered} \rm\bullet \: \: \: \sin( \alpha + \beta ) = (1)/(3) \: \: \: \: \: and \: \: \: \: \: \cos( \alpha - \beta ) = (2)/(3) \\ \end{gathered}`

Now,

`\begin{gathered} y= \bigg( ( \sin( \alpha ) )/( \cos( \beta ) ) + ( \cos( \beta ) )/( \sin( \alpha ) ) + ( \cos( \alpha ) )/( \sin( \beta ) ) + ( \sin( \beta ) )/( \cos( \alpha ) ) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \bigg( ( \sin( \alpha ) )/( \cos( \beta ) ) + ( \sin( \beta ) )/( \cos( \alpha ) ) + ( \cos( \beta ) )/( \sin( \alpha ) ) + ( \cos( \alpha ) )/( \sin( \beta ) ) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \bigg( ( \sin( \alpha ) \cos( \alpha ) + \sin( \beta \cos( \beta ) ) )/( \cos( \beta ) \cos( \alpha) ) + ( \sin( \alpha) \cos( \alpha ) + \sin( \beta ) \cos( \beta ) )/( \sin( \alpha ) \sin( \beta ) ) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \bigg( ( \sin( \alpha + \beta ) )/( \cos( \beta ) \cos( \alpha) ) + ( \sin( \alpha + \beta ) )/( \sin( \alpha ) \sin( \beta ) ) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \sin^(2) ( \alpha + \beta ) \bigg( ( 1 )/( \cos( \beta ) \cos( \alpha) ) + ( 1 )/( \sin( \alpha ) \sin( \beta ) ) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \sin^(2) ( \alpha + \beta ) \bigg( (\cos( \beta ) \cos( \alpha) + \sin( \alpha ) \sin( \beta ) )/( \cos( \beta ) \cos( \alpha) \sin( \alpha ) \sin( \beta )) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \sin^(2) ( \alpha + \beta ) \bigg( (\cos( \alpha - \beta ) )/( \cos( \beta ) \cos( \alpha) \sin( \alpha ) \sin( \beta )) \bigg)^(2) \\\end{gathered}`

`\begin{gathered} \implies y= \frac{4\sin^(2) ( \alpha + \beta ) \cdot\cos^(2) ( \alpha - \beta ) }{ \left \{2\cos( \alpha ) \cos( \beta ) \cdot 2\sin( \alpha ) \sin( \beta ) \right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= \frac{4\sin^(2) ( \alpha + \beta ) \cdot\cos^(2) ( \alpha - \beta ) }{ \left \{\cos( \alpha + \beta ) + \cos( \alpha + \beta ) \right \} ^(2) \left \{ \cos( \alpha - \beta ) - \cos( \alpha + \beta ) \right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= \frac{4\sin^(2) ( \alpha + \beta ) \cdot\cos^(2) ( \alpha - \beta ) }{ \left \{ \cos^(2) ( \alpha - \beta ) - \cos^(2) ( \alpha + \beta ) \right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= \frac{4\sin^(2) ( \alpha + \beta ) \cdot\cos^(2) ( \alpha - \beta ) }{ \left \{ \cos^(2) ( \alpha - \beta ) - 1 + \sin^(2) ( \alpha + \beta ) \right \}^(2) } \\\end{gathered}`

Putting the values given above, we get,

`\begin{gathered} \implies y= \frac{4 \cdot (1)/(9) \cdot(4)/(9) }{ \left \{ (4)/(9) - 1 + (1)/(9) \right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= \frac{(16)/(81) }{ \left \{ (5)/(9) - 1\right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= \frac{(16)/(81) }{ \left \{ (5 - 9)/(9)\right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= \frac{(16)/(81) }{ \left \{ (- 4)/(9)\right \}^(2) } \\\end{gathered}`

`\begin{gathered} \implies y= ((16)/(81) )/( (16)/(81)) \\\end{gathered}`

`⟹y=1`