### Prove the identity, please.

Posted:

**Tue 07 June, 2011 18:46**Prove the following identity.

cscθ - cosθ cotθ = sinθ

cscθ - cosθ cotθ = sinθ

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Posted: **Tue 07 June, 2011 18:46**

Prove the following identity.

cscθ - cosθ cotθ = sinθ

cscθ - cosθ cotθ = sinθ

Posted: **Tue 07 June, 2011 18:56**

The best way to approach these kinds of problems, is to almost always, convert the `csc(x) to 1/sin(x), sec(x) to 1/cos(x), and cot(x) to cos(x)/sin(x).`

Left Hand Side = LHS:

`1/sin(theta) - cos(theta)(cos(theta)/sin(theta))=`

`1/sin(theta)-(cos^2(theta))/sin(theta)=` Common denominator is `sin(theta)`

`(1- cos^2(theta))/sin(theta)=` now using the `sin^2(theta) + cos^2(theta) = 1` we can sub `1 - cos^2(theta) = sin^2(theta)`

`(sin^2(theta))/sin(theta)=` now, you can cancel one sin from the top with the one at the bottom and you are left with only:

`sin(theta) = Right hand side`

Left Hand Side = LHS:

`1/sin(theta) - cos(theta)(cos(theta)/sin(theta))=`

`1/sin(theta)-(cos^2(theta))/sin(theta)=` Common denominator is `sin(theta)`

`(1- cos^2(theta))/sin(theta)=` now using the `sin^2(theta) + cos^2(theta) = 1` we can sub `1 - cos^2(theta) = sin^2(theta)`

`(sin^2(theta))/sin(theta)=` now, you can cancel one sin from the top with the one at the bottom and you are left with only:

`sin(theta) = Right hand side`