Given \tan \theta = -\frac{4}{3}, where \frac{\pi}{2} < \theta < \pi, what is the value of \frac{1}{\sin \theta + \cos \theta}?

Answer 1

I will guess that you meant:

tanØ = 4/3, π/2 < Ø < π , or Ø is in quadrant II
construct the right-angled triangle, and you should recognize the 3-4-5 triangle.

guessing that you want:
1/sinØ + cosØ

in II, the sine is + , but the cosine is -
sinØ = 4/5 and cosØ = -3/5

1/sinØ + cosØ
= 1/(4/5) + (-3/5)
= 5/4 - 3/5
= 13/20