Hello all,
In our math class, we are practicing the trigonometric identities (i.e., sin^2(x)+cos^2(x)=1 or cot(x)=cos(x)/sin(x). Now, we are working on proofs that two sides of an equation are equal (for example, sin(x)*csc(x)=1; sin(x)csc(x)=sin(x)/sin(x)=1; 1=1). I am not having much trouble with these problems, but two are proving rather difficult for me. They are:
csc^2(x)/cot(x)=csc(x)sec(x)
and
cot^3(x)/csc(x)=cos(x)*csc^2(x)-1
Could I please get a kick in the right direction?
Thanks,
Timothy
1)
csc²(x)=1/sin²(x) and cot(x)=cos(x)/sin(x)
=> csc²(x)/cot(x) = sin(x)/(sin²(x)*cos(x))
We can scrap one of the sinus factors in both the numerator and denominator, so that the expression equals:
1/(sin(x)*cos(x))
Since csc(x)=1/sin(x) and sec(x)=1/cos(x)
the expression we found equals csc(x)*sec(x). So we have proven the equality.
2)
csc(x)=1/sin(x) and cot(x)=cos(x)/sin(x)
=> cot^3(x)/csc(x)=cos^3(x)*sin(x)/sin^3(x)
=cos^3(x)/sin²(x)
Now, we know that cos²(x)=1-sin²(x)
=> cos^3(x)=cos(x)*(1-sin²(x))=cos(x)-cos(x)*sin²(x)
So, when we fill this in in the equation we get that:
cos^3(x)/sin²(x)=(cos(x)-cos(x)*sin²(x))/sin²(x)
We can split this fraction in two separate fractions, namely:
(cos(x)/sin²(x)) - (cos(x)*sin²(x)/sin²(x))
= cos(x)*csc²(x) - cos(x)
= cos(x) (csc²(x)-1)
It seems to me you forget a pair of brackets in your question.
Thank you very much for the assistance. I understand this perfectly. Have a good day. :)
Hi Timothy,
I'd be happy to help you with these proofs. Let's take them one at a time.
To prove the equation csc^2(x)/cot(x) = csc(x)sec(x), we can start by working on the left side of the equation and simplify it step by step until we reach the right side.
First, let's rewrite csc^2(x) as (1/sin^2(x)). Now we have:
(1/sin^2(x))/cot(x)
Next, let's rewrite cot(x) as (cos(x)/sin(x)). Now we have:
(1/sin^2(x))/(cos(x)/sin(x))
To divide by a fraction, we can multiply by its reciprocal. So, we can rewrite the equation as:
(1/sin^2(x)) * (sin(x)/cos(x))
Now, let's simplify the expression by canceling out common terms:
1 * sin(x) / (sin^2(x) * cos(x))
Now, we can write sin(x) as (1/csc(x)). In other words, sin(x) = 1/csc(x):
(1/csc(x)) / (sin^2(x) * cos(x))
Next, let's rewrite sin^2(x) as (1 - cos^2(x)). This is based on the Pythagorean Identity sin^2(x) + cos^2(x) = 1:
(1/csc(x)) / ((1 - cos^2(x)) * cos(x))
Now, let's multiply the numerators and denominators together:
1 / (csc(x) * (1 - cos^2(x)) * cos(x))
Finally, we can rewrite csc(x) as (1/sin(x)) and cos(x) as (1/sec(x)). This gives us:
1 / ((1/sin(x)) * (1 - cos^2(x)) * (1/sec(x)))
Now, let's simplify the expression further:
1 / ((1/sin(x)) * (1 - cos^2(x)) * (1/(1/cos(x))))
At this point, you should notice that the terms in the denominator cancel out, leaving you with:
1 / (1 - cos^2(x))
Now, recall the Pythagorean Identity cos^2(x) + sin^2(x) = 1. By rearranging this identity, we can write cos^2(x) as (1 - sin^2(x)):
1 / (1 - (1 - sin^2(x)))
Now, simplify the expression:
1 / sin^2(x)
Finally, recall that csc(x) is the reciprocal of sin(x). So, we can write sin^2(x) as (1/csc^2(x)):
1 / (1/csc^2(x))
This gives us:
csc^2(x)
And this is equal to the right side of the equation we wanted to prove. Therefore, we have successfully proved that csc^2(x)/cot(x) = csc(x)sec(x).
I hope this explanation helps you understand the steps required to prove this equation. Let me know if you have any further questions or need help with the second proof.