Çözüldü Pythagorean Theorem - Inverse Trigonometric Function

A slightly hardened version of a simple SAT practice problem:

If a woman has a 13-meter-long ladder and sets the base of the ladder on ground level 5 meters from the side of a house, what is the vertical distance between the top of the ladder and the line horizontally drawn from the half of the ladder to the house?

A) (2 / 5)·cos[ arctan(5 / 13) ]
B) (5 / 2)·tan[ arcsin(5 / 13) ]
C) (13 / 5)·cot[ arcsin(5 / 13) ]
D) 6.5·sin[ arctan(12 / 5) ]
E) 5.5·sec[ arcsin(12 / 13) ]

Applying the Pythagorean Theorem, the Height of the house: (13^2 - 5^2)^0,5 = 12 meters
The acute angle between the ladder and the line horizontally drawn from the midpoint of the ladder: arctan(12 / 5)
Length of the upper half of the ladder: 13 / 2 = 6.5 meters
The vertical distance between the top of the ladder and the line horizontally drawn from the half of the ladder to the house:
6.5·sin[ arctan(12 / 5) ].

Original Question and Solution In The Book:

https://i.ibb.co/rmKrRnd/Ladder-Question-Sol.png
https://dynamicpapers.com/wp-content/uploads/2019/12/Acing-the-SAT-Common-Exam.pdf
Page 155 (163 in the pdf file), Question 26
Solution: Page 167 (175 in the pdf file)