I've just bought my first defender. When I apply the handbrake both the handbrake warning light and brake warning light comes on. Is this normal or should it just be the handbrake warning light

What type/year of defender do you have ?
Can you post a picture of the dash ?

The defender is 1994. Won't be able to get photo of dash until tomorrow.

Prolly brake fluid is low or your brake lights are earthing through the hand brake cable ...

Just fixed the handbrake warning light switch on my very early (3rd Jan!) 1995 Defender 90 300 Tdi, light never worked when I brought it, spent 30 mins taking the switch off the handbrake, pried it apart, 'adjusted' the little copper switch bar, a spray of switch cleaner, job done.
Anyway like Hippy Jim, when my handbrake is applied, both the handbrake and brake level warning light come on. Fluid level is fine.
Applied RTFM theory and checked the original handbook for a description of its function and helpfully it skips over the handbrake warning light when listing the functions of all the warning lights. Also the Defender 300 Tdi workshop manual doesn't have a handbrake warning circuit in the diagram, so no help there either.
For now just assuming that it is 'working as designed' and will check the brake fluid level warning light ASAP.

It could be when hand brake on brake fluid lamp lights as a check it is working