Prove that:
i) `sin40^(@)cos50^(@)+cos40^(@)sin50^(@)=1`
ii) `cot(270^(@)-theta)cot(270^(@)+theta)(cot(540^(@)-theta)cot(540^(@)+theta)=1`
iii) `cospi/8+cos(3pi)/8+cos(5pi)/8+cos(7pi)/8=0`

Explanation

Let's solve the given problems step by step.

### Part i: Prove that \( \sin 40^\circ \cos 50^\circ + \cos 40^\circ \sin 50^\circ = 1 \)

**Step 1: Use the sine addition formula.**
The expression \( \sin A \cos B + \cos A \sin B \) can be rewritten using the sine addition formula:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
...