Step-by-step explanation:
(sec(8A) − 1) / (sec(4A) − 1)
Multiply by the reciprocal:
(sec(8A) − 1) / (sec(4A) − 1) × (sec(4A) + 1) / (sec(4A) + 1)
(sec(8A) − 1) (sec(4A) + 1) / (sec²(4A) − 1)
(sec(8A) − 1) (sec(4A) + 1) / tan²(4A)
According to double angle formula, sec(2x) = sec² x / (2 − sec² x). So we can rewrite the first term as:
sec(8A) − 1
sec² (4A) / (2 − sec² (4A)) − 1
sec² (4A) / (2 − sec² (4A)) − (2 − sec² (4A)) / (2 − sec² (4A))
(sec² (4A) − 2 + sec² (4A)) / (2 − sec² (4A))
(2 sec² (4A) − 2) / (2 − sec² (4A))
2 (sec² (4A) − 1) / (2 − sec² (4A))
2 tan² (4A) / (2 − sec² (4A))
Substituting:
2 (sec(4A) + 1) / (2 − sec² (4A))
Rearranging:
2 (sec(4A) + 1) / (1 + 1 − sec² (4A))
2 (sec(4A) + 1) / (1 − tan² (4A))
From half angle formula, we know tan(x/2) = tan x / (1 + sec x). So we can rewrite the numerator as:
sec(4A) + 1
tan(4A) / tan(2A)
Substituting:
2 tan(4A) / (tan(2A) (1 − tan² (4A)))
Using double angle formula:
tan(8A) / tan(2A)
