Here are the steps to find the value of sin theta + (20 / cos theta) * (1 - cos theta) when tan theta = 8/15:
Step 1: Begin by finding the value of sin theta and cos theta using the given tan theta.
We know that tan theta = opposite/adjacent.
Given tan theta = 8/15, we can assign opposite = 8 and adjacent = 15.
Using the Pythagorean theorem, we can find the hypotenuse as follows:


Therefore, the hypotenuse is 17.
Using the definitions of sine and cosine:


So, we have sin theta = 8/17 and cos theta = 15/17.
Step 2: Substitute the values of sin theta and cos theta into the given expression:

Substituting sin theta = 8/17 and cos theta = 15/17:

Simplifying further:



Simplifying further by finding the common denominator:





Therefore, the value of the expression is 24/17.