What is i^90 + i^43???????

-1 - i

So that you can follow what I do, I'll so this the long way.

1 * i = i
i * i = i^2 = -1
i * i * i = i^3 = -i
i * i * i * i = i^4 = -1 * -1 = 1

i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1

So you can see a pattern here. Now
i^90 = (i^5)^18 = i^18 = i^3 * i^6 = -i * 1 = -i


i^43 = i^40 * i^3 = (i^5)^8 * i^3 = i^8 * i^3 = 1 * -i = -i

So,
i^90 + i^43 = -i + -i = -2i

Depends doesn't it. What's i?

Anyhow, if you mean i to be defined by i^2 = -1; then your answer is -1 - i = - (1 + i).

That results because any even exponent of i = -1 and any odd exponent of i = -i.