Question

Cot theta +Csc theta - 1/cot theta - csc theta +1=?

Answer 1

I get `1`, see below.

Explanation:

We seek:

`x=\cot \theta + \csc \theta + {1}/{\cot \theta - \csc \theta} + 1`

Plug in `\csc \theta={1}/{\sin \theta}`, `\cot \theta={\cos \theta}/{\sin \theta}`, then:

`x={\cos \theta}/{\sin \theta} + {1}/{\sin \theta} + {1}/{{\cos \theta}/{\sin \theta} - {1}/{\sin \theta}} + 1`

`x={\cos \theta+1}/{\sin \theta} + {\sin \theta}/{\cos - 1} + 1`

`x={\cos^2 \theta-1+\sin^2 \ theta}/{\sin \theta(\cos - 1)} + 1`

But `\sin^2 \theta + \cos^2 \theta =1` so the fraction cancels leaving just the constant `1`.