Answer
$3a+2b-2c$
Work Step by Step
Note that the angles $t+1000\pi$ and $t - 1000\pi$ are both coterminal with the angle $t$ since $1000\pi$ is a multiple of $2\pi$.
Note further that coterminal angles have the same value for the trigonometric functions.
Thus, the given expression is equivalent to
$=\cos{t}+\cos{t}-\tan{t}-\tan{(t+999\pi)}-\sin{t}+4\sin{t}$
Recall that the period of the tangent function is $\pi$. This means that the angles $t$ and $t+999\pi$ have the same value as they are 999 periods apart.
Thus, the expression above is equivalent to:
$=\cos{t}+\cos{t}-\tan{t}-\tan{t}-\sin{t}+4\sin{t}$
Use the given to obtain:
$=b+b-c-c-a+4a
\\=3a+2b-2c$
