More Trig Identities!

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!

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