Simplifying Sine and Cosine: Is this done right?

Question

Simplify the following expression:

sin^2(x)-1 / cos(x)

------------------------------
Here's my answer:

[1 - cos(2x)]/2
---------------------
cos(x)

I'm not really sure how to simplify pass this part. Help needed.

Answer 1

The expression sin^2(x) actually means (sin x)^2. In the background, keep in mind that this actually means (sin x)* (sin x). This is part of the Pythagorean trig identity...
(sin x)^2 + (cos x)^2 = 1

One of the strategies in doing all of these rewriting trig problems is to try to get everything in terms of sine or everything in terms of cosine. In this case the cosine in the denominator is a clue to rewrite everything with cosine... The Pythagorean trig identity helps to rewrite things (as does the double angle and half-angle formulas but sometimes it makes it more confusing - typically you'll use those only as a way to get rid of angles that look like 2x or 1/2x)...

Let's use that identity to rewrite the numerator in terms of cosine. It's a matching game. Get your numerator to look like something else, given that you start with the Pythagorean trig identity. Well, we know that sine and cosine are not in the numerator. Let's separate them by subtracting (cos x)^2 from each side...
(sin x)^2 = 1 - (cos x)^2

Now in the numerator of the problem, there was a 1 with the sine, let's move the 1 to the other side by subtracting 1 from each side...
(sin x)^2 - 1 = - (cos x)^2

There's your numerator rewritten as - (cos x)^2. We'll use this equality to replace the numerator.

So your problem...
((sin x)^2 - 1 ) / cos x

Replace with the equality we found from the identity...
-(cos x)^2 / cos x

Remember that (cos x)^2 actually means (cos x)*(cos x). So, you can canel one of those cosines on top with the one on the bottom and you're left with..
- cos x

Answer 2

You used a double-angle identity rather than the Pythagorean identity. As you can see from the other answers sin^2(x) + cos^2(x) = 1 makes the simplification possible.

Answer 3

Start with sin^2 (x) + Cos^2 (x) =1
So we can replace sin^2 (x) with 1-Cos^2 (x)
Doing so, we have - Cos^2 (x)/Cos x = -Cos x
I think you confused a cos^2 (x) for a cos (2x)