If 5 +12 cot θ = cosec θ and 12 - 5cot θ = Kcosec θ, then find K² ?

168

5 + 12cot θ = cosec θ            12 - 5 cot θ = Kcosec θ

square both side                     

5² + 144cot² θ + 120cot θ= cosec² θ  ....(i)

144 + 25cot² θ - 120cot θ = K²cosec² θ  .... (ii)

add (i) and (ii)

(5² + 12²) + (12² + 5² ) cot² θ = cosec² θ (1+ K²)

13² + 13²cot² θ = cosec² θ (1+ K²)

13² (1+ cot² θ) = cosec² θ (1+ K²)

169 cosec² θ = cosec² θ (1+ K²)

K² = 168