Find the exact value of secθ if cscθ=-4/3 and the terminal side of θ lies in quadrant iii.

a) sec θ = 4/5
b) sec θ = -(4 sqrt 7)/7
c) sec θ = 5/4
d) sec θ = (sqrt 7)/4

Quadrant is III so, sign would be negative for Sec.
Now, we know, csc = H / P
So, H = 4, P = 3
Calculate for B, 
B² = H² - P²
B² = 4² - 3²
B² = 16 - 9
B = √7

We know, sec Ф = H / B = 4 / √7
We can write it as: -4√7 / 7  [ -ve sign for 3rd quadrant ]

In short, Your Answer would be Option B

Hope this helps!