Trigonometry
posted by Timothy on .
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)=1sin²(x)
=> cos^3(x)=cos(x)*(1sin²(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. :)