Answer:
Therefore,
(sin theta + cos theta) / [cos theta (1 - cos theta)] = sec theta + csc theta
= [sqrt(1 - cos^2 theta) + cos theta] / [cos theta (1 - cos theta)]
Explanation:
We know that:
tan theta = sin theta / cos theta
Using this, we can say:
8/15 = sin theta / cos theta
We also know that:
sin^2 theta + cos^2 theta = 1
Rearranging, we get:
sin^2 theta = 1 - cos^2 theta
Taking the square root, we get:
sin theta = sqrt(1 - cos^2 theta)
Putting the value of sin theta in terms of cos theta in the given expression, we get:
(sin theta + cos theta) / [cos theta (1 - cos theta)]
= [sqrt(1 - cos^2 theta) + cos theta] / [cos theta (1 - cos theta)]
= [sqrt(1 - cos^2 theta) / cos theta] + 1 / (1 - cos theta)
= sec theta + csc theta
Now, we need to find the value of sec theta and csc theta in terms of cos theta. We know that:
sec theta = 1 / cos theta
csc theta = 1 / sin theta
= 1 / sqrt(1 - cos^2 theta)
Substituting these values in the expression we derived earlier, we get:
(sec theta + csc theta) = 1/cos theta + 1/ sqrt(1 - cos^2 theta)
= [sqrt(1 - cos^2 theta) + cos theta] / [cos theta (1 - cos theta)]