# Thread: Trig Expressions and Identities.

1. ## Trig Expressions and Identities.

1. Mulitply and simplify-
cosXsinX(secX-cscX)
I got 1 by turning it into 1 but Im not 100% sure.

2. Simplify expression-
(15tanXcscX-3cscX)/(9tanXcscX-3cscX)

3. Establish the Indentity
(1-sin2X)/(sinX-cscX)=-sinX

4. Establish the identity
cotX(cotX+tanX)=csc^2X

5. Establish the indentity
1- (cos^2X/1+sinX)=sinX

6. (sinX)/(1+cosX)+(1+cosX)/(sinX)= 2cscX

I've been working on this forever and cant seem to get these answers, any help is awesome! THANKS SO MUCH

2. For the first one, I'm not sure how you got 1 hehe

cos(x)sin(x) (sec(x)-csc(x))

(cos(x)sin(x)/cos(x)) - (cos(x)sin(x)/sin(x))

sin(x) - cos(x)

and for the fourth one..

Cot(x) (cot(x)+tan(x)) = csc^2(x)

multiply out..

cot^2(x) + (cot(x)tan(x)) = csc^2(x)

cot^2(x) + 1 = csc^2(x)

which is a common trig identity

and for the last one...

make a common denominator, and remember that sin^2(x) + cos^2(x) = 1. then just do some algebra magic

3. Originally Posted by amanda0603

1. Mulitply and simplify-
cosXsinX(secX-cscX)
I got 1 by turning it into 1 but Im not 100% sure.
$\displaystyle sec(x)-csc(x) = \frac{1}{cos(x)} - \frac{1}{sin(x)}$. Get the same denominator which will then cancel with the $\displaystyle cos(x)sin(x)$ also present

$\displaystyle \frac{1}{cos(x)} - \frac{1}{sin(x)} = \frac{sin(x)-cos(x)}{cos(x)sin(x)}$ so the final answer is

$\displaystyle \frac{cos(x)sin(x)(sin(x)-cos(x)}{cos(x)sin(x)} = sin(x)-cos(x)$

2. Simplify expression-
(15tanXcscX-3cscX)/(9tanXcscX-3cscX)
Factor and cancel:
$\displaystyle \frac{3csc(x)(5tan(x)-1)}{3csc(x)(3tan(x)-1)} = \frac{5tan(x)-1}{3csc(x)-1}$

3. Establish the Indentity
(1-sin2X)/(sinX-cscX)=-sinX
4. Establish the identity
cotX(cotX+tanX)=csc^2X
$\displaystyle cot(x)+tan(x) = \frac{1+tan^2(x)}{tan(x)} = \frac{sec^2(x)}{tan(x)}$

$\displaystyle \frac{cot(x)sec^2(x)}{tan(x)} = \frac{cos^2(x)sec^2(x)}{sin^2(x)} = csc^2(x)$

5. Establish the indentity
1- (cos^2X/1+sinX)=sinX
$\displaystyle \frac{cos^2(x)}{1+sin(x)} = \frac{1-sin^2(x)}{1+sin(x)}$

use the difference of two squares on the numerator and 1+sin(x) should cancel