Find the number of 2-digit positive integers N such that tanN15° + cot 15° is an integer.
Answers
Given : tan (N*15°) + cot ( N*15°) is an integer
To Find : number of 2-digit positive integers N
Solution:
tan (N*15°) + cot ( N*15°)
= Sin (N*15°)/Cos (N *15°) + Cos (N *15°) /Sin (N*15°)
= ( Sin² (N*15°) + Cos² (N *15°) )/ Sin (N*15°) Cos (N *15°)
= 1/ Sin (N*15°) Cos (N *15°)
= 2/ 2 Sin (N*15°) Cos (N *15°)
= 2/ Sin (N * 30°)
Sin (N * 30°) =±1 or Sin (N * 30°) = ±1/2 will give an integer
Sin (N * 30°) = Sin (180°n + 90° )
=> N * 30° =180°n + 90°
=> N = 6n + 3
k from 2 to 16 will given N = 15 to 99 Hence 15 such numbers
N = 15 , 21 ,27 , 33 , 39 , 45 , 51 , 57 ,63 , 69 , 75 , 81 ,87 , 93 ,99
Sin (N * 30°) =±1/2
N * 30° = n*180° ± 30°
=> N = 6n ± 1
N = 6n + 1 n = 2 to 16 N from 13 to 97 Hence 15 such numbers
N = 13 , 19, 25 , 31 , 37 ,43 , 49 , 55 ,61 , 67 , 73 , 79 , 85 , 91 , 97
N = 6n - 1 n = 2 to 16 N from 11 to 95 Hence 15 such numbers
N = 11, 17 , 23, 29 ,35 , 41 , 47 , 53 , 59, 65 , 71 , 77 , 83 , 89 , 95
Hence 15 + 15 + 15 = 45 Such 2 Digits numbers are possible
where tan (N*15°) + cot ( N*15°) is integer