Question

Need some help, lots actually, with Trigonometric Identities.

\[\frac{ \sec \Theta }{ \csc \Theta - \cot \Theta} - \frac{ \sec \Theta }{ \csc \Theta + \cot \Theta } = 2 \csc \Theta\]

Have no idea where to start.

Answer 1

is that right for the left side?

Answer 2

Looks right, Here is what it is
\[\frac{ \sec \Theta }{ \csc \Theta - \cot \Theta} - \frac{ \sec \Theta }{ \csc \Theta + \cot \Theta } = 2 \csc \Theta\]

Answer 3

oh ok i made a mistake in first fraction

Answer 4

hold on - let me rethink this one

Answer 5

Sure, no problem.
I have a list here that says
\[\csc \Theta = \frac{ 1 }{ \sin \Theta }, \sec \Theta = \frac{ 1 }{ \cos \Theta }, \cot \Theta = \frac{ 1 }{ \tan \Theta }\] if that helps at all. Just not sure how to make them work.

Answer 6

subtractingthe 2 fractions on LHS we get

secO(cscO + cotO) - secO(cscO - cotO)
---------------------------------
csc^2 O - cot^2 O

the denominator = 1

so we have

Answer 7

secO(cscO + cotO) - secO(cscO - cotO)

Answer 8

simplifying this we get
2 sec O cot O

Answer 9

I think I left out something in the original post. I am suppose to verify it.

Answer 10

2 sec O cot O = 2 cos O 2
--- * ----- = --- = 2 csc O = RHS
cos O sin O sin O

Answer 11

- there yougo - this verifies it

Answer 12

yes when verifying you need to use known identities to convert the more complex side of the identity to get the other side

(LHS - left hand side, RHS = right hand side)

Answer 13

did you follow that ok ?

Answer 14

Not really, Trying to figure it out but i'm just not getting it

Answer 15

there's a known identity csc^ x = 1 + cot^x
when we rearrange this it becomes csc^2 x - cot^x = 1

i used that one to simplify the fraction

Answer 16

I slso used sec x = 1/ cos x and cot x = cos x / sin x

Answer 17

subtracting the fractions was just algebra

Answer 18

I'm sorry just not understanding it for some reason. It's probably really simple once it's understood.

Answer 19

yes
do you see how i got csc^2 O - cot^2 O as the denominator?

Answer 20

its simply the product of csc O - cot O and csc o + cot O

Answer 21

like ( a + b)( a - b) = a^2 - b^2

Answer 22

anyway sorry but i gotta go right now
good luck