Is there an easy way to remember the special 8 trig. identities

For the first identity, you just have to remember the trig functions in terms of sin and cos:

tanx = sinx/cosx
cotx = cosx/sinx
secx = 1/cosx
cscx = 1/sinx

Then you get:
(secx)(cosx) = (1/cosx)(cosx) = cosx/cosx = 1
(cscx)(sinx) = (1/sinx)(sinx) = sinx/sinx = 1
(tanx)(cotx) = (sinx/cosx)(cosx/sinx) = (sinx/sinx)(cosx/cosx) = 1

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The second identity you show is wrong. You should have sec²x, and the +1 should be with tan²x

The identities tan²x + 1 = sec²x and cot²x + 1 = csc²x
follow directly from sin²x + cos²x = 1

So all you have to remember is sin²x + cos²x = 1 and you can derive the other two as follows:

sin²x + cos²x = 1 ......... divide both sides by cos²x
sin²x/cos²x + cos²x/cos²x = 1/cos²x
tan²x + 1 = sec²x

sin²x + cos²x = 1 ......... divide both sides by sin²x
sin²x/sin²x + cos²x/sin²x = 1/sin²x
1 + cot²x = csc²x

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When you are trying to solve other trig identities, usually the simplest way is to express everything in terms of sinx and cosx

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