Question

can somebody please help me with this question? i am completely lost.

Use basic identities to simplify the expression. (picture below)

Answer 1

\(\large\color{black}{ \displaystyle \frac{ \cos^2\theta}{\sin^2\theta}+\csc\theta\sin\theta }\)

Answer 2

some basic identities:

\(\large \color{black}{\sin\theta\csc\theta=1}\)
\(\large \color{black}{\cos\theta\sec\theta=1}\)
\(\large \color{black}{\tan\theta\cot\theta=1}\)

Answer 3

i still want to ask, do you have any options?

Answer 4

these are my options

Answer 5

ok, then you need 2 more rules"

\(\color{black}{\large \dfrac{\sin\theta}{\cos\theta}=\tan\theta}\)

\(\large \color{black}{\tan^2\theta+1=\sec^2\theta}\)

Answer 6

well, the first rule also applies when sine and cosine are raised to any power of n, such that:

\(\color{black}{\large \dfrac{\sin^n\theta}{\cos^n\theta}=\tan^n\theta}\) (for all n, of course)

Answer 7

or maybe u need 2 More rules :3 Mo\[\huge\rm \frac{ \cos \theta }{ \sin \theta}= \cot \theta \]
\[\huge\rm cot^2 \theta +1=\csc^2\theta \]

Answer 8

oh, Nnesha, you are right, because it is cosine/sine, not the other way around as I thought for some reason:)

and thus, the 2nd rule you posted comes about.

Answer 9

so, what exactly do i do with the rules you guys gave me?

Answer 10

\(\LARGE \color{blue}{\unicode{x10009999999}}\)\(\LARGE \color{blue}{\unicode{x120D999}}\)\(\LARGE \color{blue}{\unicode{x3D5}}\)\(\LARGE \color{blue}{\unicode{x8594}}\) \(\LARGE \color{blue}{:)}\)

Answer 11

u apply them to your problem

Answer 12

\(\large\color{black}{ \displaystyle \frac{ \cos^2\theta}{\sin^2\theta}+\csc\theta\sin\theta }\)

apply to the second term, one of the rules that I posted in the very beginning.

Answer 13

okay

what i think i understand is
cos^2(theta) / sin^2(theta) = 1

Answer 14

then cos/sin = cot

Answer 15

would it then be cot^2(theta) = 1 ?

Answer 16

nope.

Answer 17

cos^2/sin^2 is not equal to one
if cos /sin =cot
then cos^2 / sin^2 = cot^2

Answer 18

cot^2 (theta) + 1 ?

Answer 19

yes right cot^2 theta +1 which is e qual to what ?

Answer 20

csc^2 (theta) ?

Answer 21

yep right

Answer 22

thank you so much!

Answer 23

my pleasure