Price elasticity of demand question

Hi, I keep going around in circles with this one.

Question: If PED = 1, indicate if the statement is T/F
“Total Expenditure will remain constant despite price changes”

I put false, BPP book said true.

The way I got False:
For PED = 1, (using an example) a 10% increase in price would cause a 10% decrease in demand.

A decrease in demand would decrease variable costs.

I can see how a PED of 0 would lead to no change in expenditure, but not PED =1.

Thanks