Question

How do you change sec^3(x) - sec(x) into an expression with only sines and cosines?

Use fundamental identities to change the expression sec^3(x) - sec(x) to one involving only sines and cosines. Then simplify.

I am at a complete loss on how I can do this. I have spent an hour looking up fundamental identities and derivatives and I still can't make any progress on this problem at all.

Answer 1

Use `sec(x)=1/cos(x)` and the Pythagorean identity `sin^2(x)+cos^2(x)=1`

Explanation:

Formatted problem: `sec^3(x)-sec(x)`

Remember that `sec(x)=1/cos(x)`.

First, we can use the distributive property to take out a `sec(x)`:
`sec(x)(sec^2(x)-1)`

Here, we consider the Pythagorean Identity `sin^2(x)+cos^2(x)=1`, which can be modified by dividing both sides by `cos^2(x)` to give us:
`sin^2(x)/cos^2(x)+cos^2(x)/cos^2(x)=1/(cos^2(x))`
which is equal to
`tan^2(x)+1=sec^2(x)`
by subtracting `1` from both sides, we get:
`tan^2(x)=sec^2(x)-1` which is most applicable for this problem.

Continuing on, we see that we can substitute `tan^2(x)=sec^2(x)-1` into the problem:
`sec(x)(sec^2(x)-1)`
`=sec(x)(tan^2(x))`

Now, since we know that `sec(x)=1/cos(x)` and `tan(x)=sin(x)/cos(x)`, we can rewrite `sec(x)(tan^2(x))` using only cosines and sines:
`1/cos(x)(sin^2(x)/cos^2(x))`

By simplifying, we get our final answer as:
`sin^2(x)/cos^3(x)`

Hope this helps.