Simplifying expressions

Question

Sec^2(x)-tan^2(-x)

Answer 1

Kim,

 

You're looking to simplify sec²(x) - tan²(-x).  In order to simplify them, the idea is usually to get everything in terms of the same function, like all secants or all tangents...or at least try to get everything in terms of sine and cosine and then see where it leads.

 

For this problem, we have an identity that turns secants into tangents, so let's try that:

 

sec²(x) = 1 + tan²(x)

 

So, we can make that replacement in the original problem, so we're down to tan.  But we're still stuck.  In order to combine functions, we need the angles to be the same, so tan²(x) and tan²(-x) don't mix yet.  However, you hopefully have some identities for dealing with negative angles?

 

In particular, tan(-x) = -tan(x).

 

So, if we have tan²(-x), that's just fancy notation for [tan(-x)]².  If we put in -tan(x) in for tan(-x), and then square it, we get two negatives multiplied together...

 

[-tan(x)]² = tan²(x)

 

Alrighty, let's put that all together.  We replace sec²(x) with 1 + tan²(x), and we replace the tan²(-x) with tan²(x).  All together:

 

1 + tan²(x) - tan²(x)

 

The tangents cancel to zero.  You're left with a grand total of...

 

1

 

Hope this helps, but let us know what questions you have about how we got there...

 

Answer 2

sec²x = 1 + tan²x

tan²(-x) = -tan²x

 

sec²x - tan²(-x)

1 + tan²x - (-tan²x)

1 + 2tan²x