sinθ = opposite/hypothenus
and
sinθ = 12/13
opposite = 12
hypothenus = 13
from pythagoras theorem, hypothenus2 = opposite2 + adjacent2
adjacent2 = hypothenus2 - opposite2
adjacent = √(hypothenus2 - opposite2)
adjacent = √(132 - 122)
adjacent = √(169 - 144)
adjacent = √(25)
adjacent = 5
And,
cosθ = adjacent/hypothenus
cosθ = 5/13
Therefore, 1 + cosθ
= 1 + 5/13
= (13 + 5)/13
= 18/13
Ans D
Another approach : Using trigonometry identity
From trigonometry identity, sin2θ + cos2θ = 1
cos2θ = 1 - sin2θ
cos2θ = 1 - (12/13)2
cos2θ = 1 - (144/169)
cos2θ = (169 - 144)/169 (After taking the LCM)
cos2θ = 25/169
cosθ = √(25/169)
cosθ = 5/13
Therefore, 1 + cosθ
= 1 + (5/13)
= 18/13