Question

How do you simplify `csc X + cot X = sin X / (1+cos X)`?

Answer 1

This is not a valid identity.

Explanation:

We can prove this is invalid by using a test value of `x=pi/4`:

`csc(pi/4)+cot(pi/4)!=sin(pi/4)/(1+cos(pi/4))`

`sqrt2+1!=(1/sqrt2)/(1+1/sqrt2)`

`sqrt2+1!=1/(sqrt2(1+1/sqrt2))`

`sqrt2+1!=1/(sqrt2+1)`

In fact, as we can might see is happening here, these functions are actually reciprocals of one another: they only intersect when their values equal `1` or `-1`.

We can also prove these are not equal by attempting to simplify the functions:

`cscx+cotx!=sinx/(1+cosx)`

`1/sinx+cosx/sinx!=sinx/(1+cosx)`

`(1+cosx)/sinx!=sinx/(1+cosx)`

Indeed, these functions are reciprocals of one another so the identity is invalid.