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

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Answer 1 · 2018-04-06T12:44:05

I get #1#, see below.

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#.