Q: Factor the expression and use the fundamental trigonometric identities to simplify:
cos²x sec²x -cos ²x =
A. cos²x cot²x
B. cos²x
C. 1
D. sin²x
A: Start with the original problem:
cos²x sec²x – cos ²x
Now, factor out a cos ²x like so:
cos²x (sec²x – 1)
Now, we have a Pythagorean Identity that says: 1 + tan²x = sec²x
Subtract 1 from both sides to get:
tan²x = sec²x – 1
So, our problem has:
cos²x (sec²x – 1)
Sub in the Pythagorean Identity like so:
cos²x (tan²x)
Now, we also know that tanx = sinx / cosx
So, tan²x = sin²x / cos²x
Sub this in and simplify:
cos²x (tan²x) = cos²x (sin²x / cos²x) = cos²x (sin²x / cos²x) = sin²x
TADA! The answer is D!