does cot (theta) + csc (theta) =1?

Answer 1

Nope
cot^2 + 1 = csc^2
sin^2 + cos^2 = 1
tan^2 + 1 = sec^2

Answer 2

No, but if you take the Phythagorean squared sides/hypotenuse relation
a^2+b^2 = c^2, where c is a hypotenuse, a is a side adjacent to a cited angle and b is the side opposite to that cited angle (it doesn't touch the angle). Now divide each term by b^2 to get,
cot^2(angle)+1= csc^2(angle)

Answer 3

cot =cos/sin
csc=1/sin so you are asking if
cos θ +1 = sin θ
the answer is no, not as a general rule.
There is a value in the 2nd quadrant for which it will be true.