Question

Please help! will fan
Verify the basic identity. What is the domain of validity?
cot theta= cos theta (csc theta)

Answer 1

@hero ?

Answer 2

@jim_thompson5910 ?

Answer 3

cotangent = cosine / sine

cosecant = 1 / sine

Think about everywhere except where the sine function is zero.

Answer 4

hmm, okay. What should i do next?

Answer 5

What did you conclude while you were thinking?

Answer 6

that if i used reciprocal identities cot theta= cot theta
that would prove the identity?

Answer 7

Are we proving the identity or discussing its Domain of Validity?

Answer 8

both

Answer 9

I see. A little better punctuation would have cleared that up.

1) Verify - Substitute \(csc(\theta) = 1/sin(\theta)\) and you'll essentlally be done with the verification, since the right-hand side will be the definition of the cotangent.

2) Okay, now is the time to think about validity. Does it work for \(\theta = 0\)?

Answer 10

no, theta can't be zero, right?

Answer 11

@tkhunny ?

Answer 12

How about \(\theta = \pi\)?

Answer 13

that can work

Answer 14

Really? What is \(\cot(\pi)\)?

Answer 15

I dont know ! D':

Answer 16

im so confused

Answer 17

You need to know that. How did you determine that \(\theta = 0\) was a problem?

Answer 18

I was told than the denominator couldn't be zero in class

Answer 19

Perfect. Where is \(\sin(\theta) = 0\)? \(\theta \in {0 + k\pi}\), where \(\pi\) is any integer.

And we have just concluded the discussion on validity. It's not valid where \(\sin(\theta) = 0\).

Answer 20

so thats 0 degress and 180 degrees?

Answer 21

And any integer multiple of \(\pi\).