SOLUTION: cot²x - csc²x = -1 for all values of x. true or false?

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Question 550824: cot²x - csc²x = -1 for all values of x.
true or false?
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
True.
For all x, cot%28x%29=cos%28x%29%2Fsin%28x%29 and csc%28x%29=1%2Fsin%28x%29
So, for all x,
cot^2x-csc^2x=%28cot%28x%29%29%5E2-%28csc%28x%29%29%5E2 = %28cos%28x%29%2Fsin%28x%29%29%5E2-%281%2Fsin%28x%29%29%5E2 = %28cos%28x%29%29%5E2%2F%28sin%28x%29%29%5E2-1%2F%28sin%28x%29%29%5E2 = %28%28cos%28x%29%29%5E2-1%29%2F%28sin%28x%29%29%5E2 = -%281-%28cos%28x%29%29%5E2%29%2F%28sin%28x%29%29%5E2 = -%28sin%28x%29%29%5E2%2F%28sin%28x%29%29%5E2 = -1

Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor's answer is wrong.  It is NOT true for ALL values
of x because the cotangent and the cosecant are undefined at all integral
multiples of 180° or pi. Let's analyze the equation for fun:

cot²x - csc²x = -1

Let's check to see if it's true for x = 60°



cot(60°) = adjacent%2Fopposite = 1%2Fsqrt%283%29
csc(60°) = hypotenuse%2Fopposite = 2%2Fsqrt%283%29

cot²x - csc²x = -1

%281%2Fsqrt%283%29%29%5E2 - %282%2Fsqrt%283%29%29%5E2 = -1

1%2F3 - 4%2F3 = -1

           -3%2F3 = -1
                  
                   -1 = -1

Yes it is, so it might be true for all values other than
where x is an integral multiple of 90° or pi%2F2  
Let's see if we can prove it as an identity:

cot²x - csc²x = -1    where x is not an integral multiple of 180° or 90°

We will use the identity 1 + cot²(t) = csc²(t) to replace csc²x

cot²x - (1 + cot²x)

cot²x - 1 - cot²x

     -1

So yes it is an identity and true for all values of x except integers
times 180°, ( or pi in radians.  These are
called quadrantal angles.)

But the answer to the initial question is "FALSE", because the left side
is not defined for those quadrantal angles.

Edwin