sec^2(x)-sec^2(x) csc^2(x)

**sec**^{2}**(x)** - **
sec**^{2}**(x)**csc^{2}(x)

Since sec^{2}(x) is a common factor among the two terms in the expression, factoring it out will yield the following:

** sec**^{2}**(x)**(1 - csc^{2}(x))

Recall the following pythagorean trig identity: 1 + cot^{2}(x) = csc^{2}(x)

subtracting csc^{2}(x) from both sides of the equation we get the following:

1 + cot^{2}(x) - csc^{2}(x) = 0

subtracting cot^{2}(x) from both sides of the equation we get the following:

**1 - csc**^{2}**(x)** =** -cot**^{2}**(x)**

We can substitute this back into the original expression after we factored out sec^{2}(x):

sec^{2}(x)(**1 - csc**^{2}**(x)**)

sec^{2}(x)(**-cot**^{2}**(x)**)

** -(sec**^{2}**(x)cot**^{2}**(x))**

Recall the following identities:

Reciprocal identity: sec(x) = 1/cos(x) ==> **sec**^{2}**(x) = 1/cos**^{2}**(x)**

Quotient identity: cot(x) = cos(x)/sin(x) ==> **cot**^{2}**(x) = cos**^{2}**(x)/sin**^{2}**(x)**

Substituting these identities into the simplified expression, we arrive at the following:

-(**sec**^{2}**(x)**)(**cot**^{2}**(x)**)

-(**1/cos**^{2}**(x)**)(**cos**^{2}**(x)/sin**^{2}**(x)**)

** -(1/sin**^{2}**(x))**

Another reciprocal identity involving sin is the following:

1/sin(x) = csc(x) ==> **1/sin**^{2}**(x) = csc**^{2}**(x)**

Therefore,

-(**1/sin**^{2}**(x)**) = -(**csc**^{2}**(x)**)

Thus,

sec^{2}(x) - sec^{2}(x)csc^{2}(x) = **
-csc**^{2}**(x)**