Is x9 in profits worth it if it takes half of my angels?

It's worth it (assuming you keep it running long enough to get your angels back, anyway) because the extra Angels aren't separate multipliers, they are addition to an existing multiplier.

What happens is your profits are:

<base value> x <N angels> x <modifier A> x <modifier B>

Doubling your angels will change it to:

<base value> x <2N angels> x <modifier A> x <modifier B>

Doubling your Angels is effectively doubling your bonus, regardless of how many Angels you had (or have). In your case, you would need another 1000 quindecillion of them just to see x2 profits.

On the other hand, if you trade in half your Angels for an extra x9 modifer, you get the following:

<base value> x <0.5N angels> x <modifier A> x <modifier B> x 9

This is a 4.5x rise in your profits. Sacrificing half your Angels cuts your bonus in half; the new total is multiplied by another 9.

There is an enourmous difference between an additive multiplier (going from x20 to x21) and multiplicative multipliers (is that a word?) (going from x20 to x20x2)

Adding more Angels is always additive; everything else is multiplicative.

So once you hit the point where you spend less than 1 - 1/(multiplier) of your Angels, you'll earn. In this case, 1 - 1/9 is about 89%, you'll be fine with spending 50% of them.

Just keep in mind you should not reset until you've earned them back, which might take a while.

I'm no Math major but the formula provided on this page should give you the correct answer.

With:

M: Multiplier
C: Cost
E_n: New Angel effectiveness
E_c: Current Angel effectiveness

The best way is to use the system they have designed in the game. If you purchase upgrades using angels, and it asks are you sure, hit no. If it does not ask, it's "Safe" or efficient enough not to cause a loss of profits. Think of it like this, if it asks 'Are you Sure?', it's because it isn't considered Safe when it consumes more than 1% of your angels. To prevent confusion, remember this doesn't mean it will never give you more profits than you currently have, but if you use their built in system, you won't ever waste angels.

This same things applies when you buy the quick buy manager (earth or mars), because with the quick buy angels, it will only buy the safe upgrades and leave any others alone.