Let = 1 sin 60 ° sin 61 ° + 1 sin 62 ° sin 63 ° + ? + 1 sin 118 ° sin 119 ° . Then the value of ( c o s e c 1 ° ) 2 is ________. [2025]

Question

Let $α = \frac{1}{\sin 60 ° \sin 61 °} + \frac{1}{\sin 62 ° \sin 63 °} + \dots + \frac{1}{\sin 118 ° \sin 119 °} .$ Then the value of ${(\frac{c o s e c 1 °}{α})}^{2}$ is ________. [2025]

Smart Achievers · Accepted Answer

(3)

$α = \sum_{r = 30}^{59} \frac{1}{\sin (2 r) ° \sin (2 r + 1) °}$

$\Rightarrow \frac{α}{cosec 1 °} = \sum_{r = 30}^{59} \frac{\sin [(2 r) ° - (2 r + 1) °]}{\sin (2 r) ° \sin (2 r + 1) °}$

$= \sum_{r = 30}^{59} (\cot (2 r) ° - \cot (2 r + 1) °)$

$= \cot 60 ° - \cot 61 ° + \cot 62 ° - \cot 63 ° + \dots + \cot 118 ° - \cot 119 °$

$\Rightarrow \frac{α}{cosec 1 °} = \cot 60 ° = \frac{1}{\sqrt{3}}$

$\Rightarrow {(\frac{cosec 1 °}{α})}^{2} = 3$

Solution

Q.
Let $α = \frac{1}{\sin 60 ° \sin 61 °} + \frac{1}{\sin 62 ° \sin 63 °} + \dots + \frac{1}{\sin 118 ° \sin 119 °} .$ Then the value of ${(\frac{c o s e c 1 °}{α})}^{2}$ is ________. [2025]

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Solution

Q. Let α=1sin60°sin61°+1sin62°sin63°+⋯+1sin118°sin119°. Then the value of (cosec1°α)2 is ________. [2025]

Ans. (3) α=∑r=30591sin(2r)°sin(2r+1)° ⇒αcosec1°=∑r=3059sin[(2r)°-(2r+1)°]sin(2r)°sin(2r+1)° =∑r=3059(cot(2r)°-cot(2r+1)°) =cot60°-cot61°+cot62°-cot63°+⋯ +cot118°-cot119° ⇒αcosec1°=cot60°=13 ⇒(cosec1°α)2=3

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Q.
Let $α = \frac{1}{\sin 60 ° \sin 61 °} + \frac{1}{\sin 62 ° \sin 63 °} + \dots + \frac{1}{\sin 118 ° \sin 119 °} .$ Then the value of ${(\frac{c o s e c 1 °}{α})}^{2}$ is ________. [2025]