Final answer:
The given equation cos A / (tan A · sin A) = 16 is simplified using trigonometric identities, leading to proving that sec A = ± √17 / 4.
Step-by-step explanation:
If the equation cos A / (tan A · sin A) equals 16, we are asked to prove that sec A equals ± √17 / 4.
First, let's simplify the given equation using trigonometric identities:
- tan A = sin A / cos A
- sec A = 1 / cos A
Substituting the identity of tan A into the equation, we get:
cos A / ((sin A / cos A) · sin A) = cos^2 A / sin^2 A
Therefore, the equation becomes:
cos^2 A / sin^2 A = 16
Which can be further written as:
(1 / sin^2 A) - 1 = 16
Simplifying, we have:
csc^2 A - 1 = 16
Knowing that csc A = 1 / sin A and using the Pythagorean identity csc^2 A - 1 = cot^2 A, we rewrite the equation:
cot^2 A = 16
Since cot A = cos A / sin A, the equation further simplifies to:
cos^2 A / sin^2 A = 16
Using another identity sin^2 A + cos^2 A = 1, we get:
cos^2 A = 16 sin^2 A
cos^2 A = 16 (1 - cos^2 A)
Which leads us to:
17 cos^2 A = 16
cos^2 A = 16 / 17
Eventually, sec A = 1 / cos A gives us:
sec A = 1 / (± √(cos^2 A))
sec A = ± √17 / 4
This proves the required result.