use trigonometric identities to simplify the expression

Question

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

Answer 1

     sec²(x) - sec²(x)csc²(x)

Since sec²(x) is a common factor among the two terms in the expression, factoring it out will yield the following:

     sec²(x)(1 - csc²(x))

Recall the following pythagorean trig identity:     1 + cot²(x) = csc²(x)

     subtracting csc²(x) from both sides of the equation we get the following:

          1 + cot²(x) - csc²(x) = 0

     subtracting cot²(x) from both sides of the equation we get the following:

          1 - csc²(x) = -cot²(x)

We can substitute this back into the original expression after we factored out sec²(x):

          sec²(x)(1 - csc²(x))

          sec²(x)(-cot²(x))

         -(sec²(x)cot²(x))

Recall the following identities:

     Reciprocal identity:     sec(x) = 1/cos(x)   ==>   sec²(x) = 1/cos²(x)

     Quotient identity:     cot(x) = cos(x)/sin(x)   ==>   cot²(x) = cos²(x)/sin²(x)

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

     -(sec²(x))(cot²(x))

     -(1/cos²(x))(cos²(x)/sin²(x))

     -(1/sin²(x))

Another reciprocal identity involving sin is the following:

     1/sin(x) = csc(x)     ==>     1/sin²(x) = csc²(x)

Therefore,

     -(1/sin²(x)) = -(csc²(x))

Thus,

          sec²(x) - sec²(x)csc²(x)  =  -csc²(x)

Answer 2

Hey there Bryce,

Trig Identities are always a fun challenge/puzzle.

Here's how I approached this problem:

First, we have sec^2(x)-sec^2(x) csc^2(x). Let's go ahead separate it just a little bit:

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

That basically shows the quantity on the right being subtracted from the quantity on the left. 

Now, let's simplify. Notice how both quantities (left and right) have a sec^2(x). So go ahead and pull that particular variable out.

[sec^2(x)] - [sec^2(x) * Csc^2(x)] becomes [Sec^2 (x)] * [1-Csc^2(x)]

Now, remember that 1+ Cot^2(x) =Csc^2(x).  

So if 1+ Cot^2(x) =Csc^2(x) and we have [Sec^2 (x)] * [1-Csc^2(x)], then we 

can substitute in the 1+Cot^2(x) in for the Csc^2(x).

That gives this: [Sec^2(x)] * [1- {1+Cot^2(x)}]. Simplify the right side of the expression:

[Sec^2(x)] *[1- {1+Cot^2(x)}] becomes Sec^2(x)* Cot^2(x).

Well, Sec(x) is 1/Cos(x) so Sec^2(x) is equal to 1/Cos^2(x).

And then remember that Cot(x) = Cos(x)/Sin(x) so that Cot^2(x) = Cos^2(x) / Sin^2(x).

So that means that Sec^2(x) * Cot^2(x) is equivalent to:

1/Cos^2(x) * [Cos^2(x) / Sin^2(x)]

The Cos^2(x) cancels out leaving you with 1/Sin^2(x), which is then simplified to Csc^2(x).

 

Hope that helped!