Let α = 1/(sin 60^° sin 61^°) + 1/(sin 62^° sin 63^°) + … + 1/(sin 118^° sin 119^°) Then the value of ((cosec 1^°)/α)² is ________.
α = 1 sin 60 ∘ sin 61 ∘ + 1 sin 62 ∘ sin 63 ∘ + ⋯ + 1 sin 118 ∘ sin 119 ∘ \alpha = \dfrac{1}{\sin 60^\circ\sin 61^\circ} + \dfrac{1}{\sin 62^\circ\sin 63^\circ} + \cdots + \dfrac{1}{\sin 118^\circ\sin 119^\circ} α = sin 6 0 ∘ sin 6 1 ∘ 1 + sin 6 2 ∘ sin 6 3 ∘ 1 + ⋯ + sin 11 8 ∘ sin 11 9 ∘ 1 terms: 1 sin ( 2 k ) sin ( 2 k + 1 ) , k = 30 , 31 , … , 59 \text{terms: } \dfrac{1}{\sin(2k)\sin(2k+1)},\quad k=30,31,\ldots,59 terms: sin ( 2 k ) sin ( 2 k + 1 ) 1 , k = 30 , 31 , … , 59
To find: ( cosec 1 ∘ α ) 2 \text{To find: } \left(\dfrac{\operatorname{cosec}1^\circ}{\alpha}\right)^2 To find: ( α cosec 1 ∘ ) 2
cot a − cot b = cos a sin b − cos b sin a sin a sin b = sin ( b − a ) sin a sin b \cot a-\cot b = \dfrac{\cos a\sin b-\cos b\sin a}{\sin a\sin b} = \dfrac{\sin(b-a)}{\sin a\sin b} cot a − cot b = sin a sin b cos a sin b − cos b sin a = sin a sin b sin ( b − a ) ⇒ 1 sin a sin b = cot a − cot b sin ( b − a ) \Rightarrow\ \dfrac{1}{\sin a\sin b} = \dfrac{\cot a-\cot b}{\sin(b-a)} ⇒ sin a sin b 1 = sin ( b − a ) cot a − cot b 1 sin x sin ( x + 1 ∘ ) = cot x − cot ( x + 1 ∘ ) sin 1 ∘ \dfrac{1}{\sin x\sin(x+1^\circ)} = \dfrac{\cot x - \cot(x+1^\circ)}{\sin 1^\circ} sin x sin ( x + 1 ∘ ) 1 = sin 1 ∘ cot x − cot ( x + 1 ∘ ) α = 1 sin 1 ∘ ∑ x ( cot x − cot ( x + 1 ∘ ) ) , x = 60 , 62 , … , 118 \alpha = \dfrac{1}{\sin1^\circ}\sum_{x}\big(\cot x-\cot(x+1^\circ)\big),\quad x=60,62,\ldots,118 α = sin 1 ∘ 1 ∑ x ( cot x − cot ( x + 1 ∘ ) ) , x = 60 , 62 , … , 118 α = 1 sin 1 ∘ [ ( cot 60 ∘ − cot 61 ∘ ) + ( cot 62 ∘ − cot 63 ∘ ) + ⋯ + ( cot 118 ∘ − cot 119 ∘ ) ] \alpha = \dfrac{1}{\sin1^\circ}\Big[(\cot60^\circ-\cot61^\circ)+(\cot62^\circ-\cot63^\circ)+\cdots+(\cot118^\circ-\cot119^\circ)\Big] α = sin 1 ∘ 1 [ ( cot 6 0 ∘ − cot 6 1 ∘ ) + ( cot 6 2 ∘ − cot 6 3 ∘ ) + ⋯ + ( cot 11 8 ∘ − cot 11 9 ∘ ) ] α = 1 sin 1 ∘ [ ∑ even x = 60 118 cot x − ∑ odd x = 61 119 cot x ] \alpha = \dfrac{1}{\sin1^\circ}\Big[\sum_{\text{even }x=60}^{118}\cot x - \sum_{\text{odd }x=61}^{119}\cot x\Big] α = sin 1 ∘ 1 [ ∑ even x = 60 118 cot x − ∑ odd x = 61 119 cot x ] cot ( 180 ∘ − θ ) = − cot θ \cot(180^\circ-\theta) = -\cot\theta cot ( 18 0 ∘ − θ ) = − cot θ ∑ odd 61 119 cot x = 0 \sum_{\text{odd }61}^{119}\cot x = 0 ∑ odd 61 119 cot x = 0 ∑ even 60 118 cot x = cot 60 ∘ + cot 90 ∘ = cot 60 ∘ \sum_{\text{even }60}^{118}\cot x = \cot60^\circ + \cot90^\circ = \cot60^\circ ∑ even 60 118 cot x = cot 6 0 ∘ + cot 9 0 ∘ = cot 6 0 ∘ cot 60 ∘ = 1 3 \cot60^\circ = \dfrac{1}{\sqrt3} cot 6 0 ∘ = 3 1 α = cot 60 ∘ sin 1 ∘ = 1 3 sin 1 ∘ \alpha = \dfrac{\cot60^\circ}{\sin1^\circ} = \dfrac{1}{\sqrt3\,\sin1^\circ} α = sin 1 ∘ cot 6 0 ∘ = 3 sin 1 ∘ 1 cosec 1 ∘ α = 1 / sin 1 ∘ 1 / ( 3 sin 1 ∘ ) = 3 \dfrac{\operatorname{cosec}1^\circ}{\alpha} = \dfrac{1/\sin1^\circ}{1/(\sqrt3\sin1^\circ)} = \sqrt3 α cosec 1 ∘ = 1/ ( 3 sin 1 ∘ ) 1/ sin 1 ∘ = 3 ( cosec 1 ∘ α ) 2 = ( 3 ) 2 = 3 \left(\dfrac{\operatorname{cosec}1^\circ}{\alpha}\right)^2 = (\sqrt3)^2 = 3 ( α cosec 1 ∘ ) 2 = ( 3 ) 2 = 3
