Prove the identity: -tan^2x + sec^2x = 1

Question

anonymous · Answer

I have no clue where to start.

anonymous · Answer

$$-\tan ^{2}x + \sec ^{2}x = 1$$

zepdrix · Answer

We have 3 important formulas that relate squares,$$\large \sin^2x+\cos^2x=1$$$$\large \tan^2x+1=\sec^2x$$$$\large \cot^2x+1=\sec^2x$$

Can we maybe use one of those identities to help us solve this?

anonymous · Answer

The second one?

zepdrix · Answer

Yah that seems like a good idea. So what we want to do to "Prove" is to leave one side alone, and try to match it with the other side.

zepdrix · Answer

$$\large -\tan ^{2}x + \color{orangered}{\sec ^{2}x} = 1$$Let's replace this sec^2x using our identity,$$\large \sec^2x=\color{orangered}{\tan^2x+1}$$

anonymous · Answer

$$-\tan^2x + \tan^2x + 1 = 1$$

anonymous · Answer

Is that right?

zepdrix · Answer

$$\large -\tan^2x+\color{orangered}{\tan^2x+1}=1$$Yes very good ^^

anonymous · Answer

Yay! So now what?

zepdrix · Answer

Hmm we have a tan^2, and a negative tan^2.. let's combine them.

anonymous · Answer

So they cancel each other out, right?
You're left with 1 = 1.

zepdrix · Answer

Yesss \c:/ yay team!

anonymous · Answer

Yay! Thank you so much! You are a life saver. :)