Can someone help me with trigonometric identities

I understand the Reciprocal and Quotient Properties for trigonometric functions but I do not fully understand the Pythagorean Property. I am fine with the cos^2x+sin^2x=1 but I am confused with the other 2 equations listed in the book. They are 1+tan^2x=sec^2x and cot^2x+1=csc^2x. The main one I do not understand is 1+tan^2x=sec^2x. Is that the same as tan^2x-sec^2x=1? Could someone explain to me how the Pythagorean Property in trigonometry works?

cos^2x+sin^2x=1
1+tan^2x=sec^2x
cot^2x+1=csc^2x

These 3 equations above are the originals and the ones that you will work with. Learn them and don't question them. They are kind of like a^2+b^2=c^2 just a formula that you can move around and work with but it is what it is.

tan^2x-sec^2x=1 does not exist.

if you want to modify any of the original 3 you move them around just like you would do r example b^2=c^2-a^2 ; same thing just looks different.

if you wanted to modify 1+tan^2x=sec^2x it would be sec^2-tan^2=1 or tan^2-sec^2= -1 or sec^2 -1=tan^2

hopefully you get the idea

you can also do what "MechE Monster" did I just though I should explain it in simpler words

It just comes from manipulation and the face that tan^2x = (sin^2x/cos^2x) and sec^2x = (1/cos^2x)

cos^2x + sin^2x = 1

divide both sides by cos^2x.....

(cos^2x/cos^2x) + (sin^2x/cos^2x) = 1/cos^2x

simplify....

1 + tan^2x = sec^2x

The other equation is derived in a similar fashion but dividing by sin^2x

1+tan^2x = 1+sin^2x / cos^2x = (sin^2x +cos^2x)/cos^2x = 1/cos^2x = sec^2x