Answer
$2a+6b+2c$
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
$=-\cos{t}+7\cos{t}+\tan{t}+\tan{(t+999\pi)}+\sin{t}+\sin{t}$
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:
$=-\cos{t}+7\cos{t}+\tan{t}+\tan{t}+\sin{t}+\sin{t}$
Use the given to obtain:
$=-b+7b+c+c+a+a
\\=2a+6b+2c$
