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.

1 Answer
May 10, 2017

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.