How do you verify ##1-tan^2x=2-sec^2x##?

2 – ##sec^2##x = 2 – ##sec^2##x

When you work on identities you have to first decide which side of the identity you want to work on. In this particular equation I decided to work on the left side to prove that it is the right side.

After you pick the side you want to work on, you have to look at it and decide if you need to do algebra or trig. In this particular problem, doing algebra first would complicate the problem, so we will do trig. In this case ##tan^2##x = ##sec^2##x – 1, and we know this from our “nifty nine” trig identities.

So using trig our problem now looks like this:
1 – ##tan^2##x = 2 – ##sec^2##x
1 – (##sec^2##x – 1)
Now after you make a change to the problem you need to decide on whether or not you will use algebra or trig to help prove the identity. In this case there is not any trig that we can do that would help, so we have to use algebra. In this case we need to distribute that unary minus sign. So now our problem looks like this:
1 – ##sec^2##x + 1
Well as you can see from this point we need to combine like terms and we will now get this:
2 – ##sec^2##x
We now check to see if there is anything else that needs to be done, but as you can see we have now proven the identity. So the final step would be to put in the = sign and all of the pieces after the = sign.
2 – ##sec^2##x = 2 – ##sec^2##x
There you go. I hope this helps.