jee-advanced 2025 Q15

jee-advanced · India · paper2 4 marks Addition & Double Angle Formulae Telescoping Sum of Trigonometric Terms
Let
$$\alpha = \frac { 1 } { \sin 60 ^ { \circ } \sin 61 ^ { \circ } } + \frac { 1 } { \sin 62 ^ { \circ } \sin 63 ^ { \circ } } + \cdots + \frac { 1 } { \sin 118 ^ { \circ } \sin 119 ^ { \circ } } .$$
Then the value of
$$\left( \frac { \operatorname { cosec } 1 ^ { \circ } } { \alpha } \right) ^ { 2 }$$
is $\_\_\_\_$.
Let

$$\alpha = \frac { 1 } { \sin 60 ^ { \circ } \sin 61 ^ { \circ } } + \frac { 1 } { \sin 62 ^ { \circ } \sin 63 ^ { \circ } } + \cdots + \frac { 1 } { \sin 118 ^ { \circ } \sin 119 ^ { \circ } } .$$

Then the value of

$$\left( \frac { \operatorname { cosec } 1 ^ { \circ } } { \alpha } \right) ^ { 2 }$$

is $\_\_\_\_$.