( s i n α c o s β + c o s α s i n β + c o s β s i n α + s i n β c o s α ) 2 {\left(\dfrac{sin\alpha}{cos\beta} + \dfrac{cos\alpha}{sin\beta} + \dfrac{cos\beta}{sin\alpha} + \dfrac{sin\beta}{cos\alpha}\right)}^{2} ( cos β s in α + s in β cos α + s in α cos β + cos α s in β ) 2
= ( cos ( α − β ) s i n β c o s β + cos ( α − β ) s i n α c o s α ) 2 = {\left(\dfrac{\cos \left(\alpha - \beta\right)}{sin\beta cos\beta} + \dfrac{\cos \left(\alpha - \beta\right)}{sin\alpha cos\alpha}\right)}^{2} = ( s in β cos β cos ( α − β ) + s in α cos α cos ( α − β ) ) 2
= ( 2 cos ( α − β ) { 1 sin 2 β + 1 sin 2 α } ) 2 = {\left(2\cos \left(\alpha - \beta\right)\left\{\dfrac{1}{\sin 2\beta} + \dfrac{1}{\sin 2\alpha}\right\}\right)}^{2} = ( 2 cos ( α − β ) { sin 2 β 1 + sin 2 α 1 } ) 2
= 16 9 ( 2 sin ( α + β ) ⋅ cos ( α − β ) sin 2 α ⋅ sin 2 β ) 2 = \dfrac{16}{9}{\left(\dfrac{2\sin \left(\alpha + \beta\right) \cdot \cos \left(\alpha - \beta\right)}{\sin 2\alpha \cdot \sin 2\beta}\right)}^{2} = 9 16 ( sin 2 α ⋅ sin 2 β 2 sin ( α + β ) ⋅ cos ( α − β ) ) 2
= 16 9 ( 4 ⋅ 1 3 ⋅ 2 3 cos ( 2 α − 2 β ) − cos ( 2 α + 2 β ) ) 2 = \dfrac{16}{9}{\left(\dfrac{4 \cdot \dfrac{1}{3} \cdot \dfrac{2}{3}}{\cos \left(2\alpha - 2\beta\right) - \cos \left(2\alpha + 2\beta\right)}\right)}^{2} = 9 16 cos ( 2 α − 2 β ) − cos ( 2 α + 2 β ) 4 ⋅ 3 1 ⋅ 3 2 2
= 16 9 ( 8 9 ( 2 cos 2 ( α − β ) − 1 ) − ( 1 − 2 sin 2 ( α + β ) ) ) 2 = \dfrac{16}{9}{\left(\dfrac{\dfrac{8}{9}}{\left(2\cos^{2}\left(\alpha - \beta\right) - 1\right) - \left(1 - 2\sin^{2}\left(\alpha + \beta\right)\right)}\right)}^{2} = 9 16 ( 2 cos 2 ( α − β ) − 1 ) − ( 1 − 2 sin 2 ( α + β ) ) 9 8 2
= 16 9 ( 8 9 8 9 − 2 + 2 9 ) 2 = \dfrac{16}{9}{\left(\dfrac{\dfrac{8}{9}}{\dfrac{8}{9} - 2 + \dfrac{2}{9}}\right)}^{2} = 9 16 9 8 − 2 + 9 2 9 8 2
= 16 9 = \dfrac{16}{9} = 9 16
i.e. \left[\dfrac{16}{9}\right] = 1
