Question

Question #538a5

Answer 1

`sin(x)/(1-cos(x))-sin(x)/(1+cos(x))=2cot(x)` is a valid identity

`sin(x)/(1-cos(x))-sin(x)/(1+cos(x))=2csc(x)` isn't a valid identity

Explanation:

The following equation

`sin(x)/(1-cos(x))-sin(x)/(1+cos(x))=2csc(x)`

Isn't a valid identity,
you can check this by substitution of an appropriate value of `x`

However, if you mean

`sin(x)/(1-cos(x))-sin(x)/(1+cos(x))=2cot(x)`

It is a valid identity

We will use

• `cos^2(x)+sin^2(x)=1`
• `cot(t)=cos(x)/sin(x)`

Look at work

`RHS=sin(x)/(1-cos(x))-sin(x)/(1+cos(x))`

`=(sin(x)(1+cos(x))-sin(x)(1-cos(x)))/((1-cos(x))(1+cos(x)))`

`=(sin(x)+sin(x)cos(x)-sin(x)+sin(x)cos(x))/(1-cos^2(x))`

`=(2sin(x)cos(x))/(sin^2(x))`

`=2cot(x)`

`=LHS`