SOLUTION: What is 1/cot( x) in terms of sin (x)?

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Question 1205934: What is 1/cot( x) in terms of sin (x)?
Found 3 solutions by MathLover1, math_tutor2020, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Use the pythagorean trig identity to determine cosine in terms of sine.
sin%5E2%28theta%29+%2B+cos%5E2%28theta%29+=+1

cos%5E2%28theta%29+=+1-sin%5E2%28theta%29

cos%28theta%29+=+%22%22%2B-sqrt%281-sin%5E2%28theta%29%29


Then,
1%2F%28cot%28theta%29%29+=+tan%28theta%29

1%2F%28cot%28theta%29%29+=+sin%28theta%29%2Fcos%28theta%29

1%2F%28cot%28theta%29%29+=+%22%22%2B-sin%28theta%29%2F%28sqrt%281+-+sin%5E2%28theta%29%29%29

Refer to this list for more info
https://en.wikipedia.org/wiki/List_of_trigonometric_identities
Scroll a bit to reach the table with the heading "Each trigonometric function in terms of each of the other five".
Look at the row that starts with and the column that has "sin" at the top.

If theta is in quadrants Q1 or Q3, then 1%2F%28cot%28theta%29%29+=+sin%28x%29%2F%28sqrt%281+-+sin%5E2%28theta%29%29%29

If theta is in quadrants Q2 or Q4, then 1%2F%28cot%28theta%29%29+=+%28-sin%28theta%29%29%2F%28sqrt%281+-+sin%5E2%28theta%29%29%29

Answer by ikleyn(52865) About Me  (Show Source):
You can put this solution on YOUR website!
.

1%2Fcot%28x%29 = sin%28x%29%2Fcos%28x%29 = +/- sin%28x%29%2Fsqrt%281-sin%5E2%28x%29%29.


In QI,    1%2Fcot%28x%29 = sin%28x%29%2Fsqrt%281-sin%5E2%28x%29%29. 


In QII,   1%2Fcot%28x%29 = -sin%28x%29%2Fsqrt%281-sin%5E2%28x%29%29. 


In QIII,  1%2Fcot%28x%29 = -sin%28x%29%2Fsqrt%281-sin%5E2%28x%29%29. 


In QIV,   1%2Fcot%28x%29 = sin%28x%29%2Fsqrt%281-sin%5E2%28x%29%29.

Solved.

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The answer by @MathLover1 is partly incomplete and partly incorrect.