Find sec^2 x (4 tan^2 x-3)
To find the value of sec^2 x (4 tan^2 x - 3), we'll break the problem down into smaller steps:
Step 1: Simplify the expression inside the parentheses.
The expression inside the parentheses is 4 tan^2 x - 3. Since tan^2 x is equivalent to (sin^2 x / cos^2 x), we can substitute it in:
4 tan^2 x - 3 = 4 (sin^2 x / cos^2 x) - 3
Step 2: Simplify the numerator.
The numerator, sin^2 x, can be simplified using the Pythagorean identity sin^2 x + cos^2 x = 1. Rearranging this equation, we get sin^2 x = 1 - cos^2 x. We can substitute this value in the numerator:
4 (1 - cos^2 x) / cos^2 x - 3
Step 3: Combine like terms.
We need to distribute the 4 across the numerator:
4 - 4 cos^2 x / cos^2 x - 3
Now, simplify the expression further:
(4 - 4 cos^2 x) / cos^2 x - 3
Step 4: Simplify the denominator.
cos^2 x can be simplified as 1 / sec^2 x by using the reciprocal identity for cosine, which states that cos^2 x = 1 / sec^2 x. Substituting this value in the denominator:
(4 - 4 cos^2 x) / (1 / sec^2 x) - 3
Simplifying further:
(4 - 4 cos^2 x) * sec^2 x - 3
Step 5: Simplify the remaining terms.
Now, we can simplify sec^2 x * (4 - 4 cos^2 x):
sec^2 x * (4 - 4 cos^2 x) = (sec^2 x * 4) - (sec^2 x * 4 cos^2 x)
= 4 sec^2 x - 4 sec^2 x cos^2 x
So, the final expression is:
4 sec^2 x - 4 sec^2 x cos^2 x