Odd Summation

Using logarithm laws, $\log _{ e }{ (\sec { \theta )-\log _{ e }{ (\csc { \theta )=\log _{ e }{ (\frac { \sec { \theta } }{ \csc { \theta } } ) } } } } }$

$=\log _{ e }{ (\frac { \sin { \theta } }{ \cos { \theta } } ) }$

$=\log _{ e }{ (\tan { \theta } ) }$

Hence the expression is equivalent to $\sum _{ \theta =1 }^{ 89 }{ \log _{ e }{ (\tan { \theta ) } } }$

$=\log _{ e }{ (\tan { 1) } +\log _{ e }{ (\tan { 2) } +...+\log _{ e }{ (\tan { 88) } +\log _{ e }{ (\tan { 89) } } } } }$

$=\log _{ e }{ (\tan { 1 } \tan { 2 } ...\tan { 88 } \tan { 89 } ) }$

Now $\tan { \theta } \tan { (90-\theta ) } \\ =\tan { \theta } \frac { \sin { (90-\theta ) } }{ \cos { (90-\theta ) } } \\ =\tan { \theta } \frac { \cos { \theta } }{ \sin { \theta } } \\ =\tan { \theta } \frac { 1 }{ \tan { \theta } } \\ =1$

Hence the expression reduces to $\log _{ e }{ (\tan { 45 } ) } \\ =\log _{ e }{ 1 } \\ =\boxed { 0 }$ .