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

##### 1 Answer

Mar 22, 2016

This is not a valid identity.

#### Explanation:

We can prove this is invalid by using a test value of

#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

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.