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

1 Answer
mason m
Mar 22, 2016

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.

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