Log of product of tangent

Relevant wiki: Properties of Logarithms - Basic

$\ln{\displaystyle\left(\prod^{89}_{x=1}{\tan{x^{\circ}}}\right)} = \ln({\tan1^o \times \tan 2^o \times \tan 3^o... \times \tan 89^o})$

Hence $\ln({\tan1^o \times \tan 2^o \times \tan 3^o... \times \tan 89^o})$ = $\ln({\tan1^o \times \tan 2^o \times \tan 3^o... \times \cot 2^o\times \cot 1^o})$

Here the turning point of tan's to cot is $\tan45^o$

Like this $\ln({\tan1^o \times \tan 2^o \times \tan 3^o... \times \tan44^{o} \times \tan 45^{o} \times \tan 46^{o} \times \tan 47^{o} \times \cot 2^o\times \cot 1^o})$

Now $\tan 46^{o}$ can be written as $\cot 44^{o}$ , $\tan 47^{o}$ can be written as $\cot 43^{o}$ and so on.. Like this

$\ln({\tan1^o \times \tan 2^o \times \tan 3^o... \times \tan44^{o} \times \tan 45^{o} \times \cot 44^{o} \times \cot 43^{o} \times ... \cot 2^o\times \cot 1^o})$

Now all the cot's will be cut by tan's resulting in

$\ln{\tan 45^{o}}$

Now $\tan 45^o = 1$ .

So $\ln{1}$ = $\boxed{0}$

$\ln{\displaystyle\left(\prod^{89}_{x=1}{\tan{x^{\circ}}}\right)} = \, \ln{\displaystyle \left( \tan{1^{\circ}} \times \tan{2^{\circ}} \times \cdots \times \tan{88^{\circ}} \times \tan{89^{\circ}}\right)}$

We know that $\tan{x^{\circ}} = \dfrac{\sin{x^{\circ}}}{\cos{x^{\circ}}}$ . Therefore, we see that

$\displaystyle \ln{\left(\tan{1^{\circ}} \times \tan{2^{\circ}} \times \cdots \times \tan{88^{\circ}} \times \tan{89^{\circ}}\right)} = \ \ln{ \left( \dfrac{\sin{1^{\circ}}}{\cos{1^{\circ}}} \times \dfrac{\sin{2^{\circ}}}{\cos{2^{\circ}}} \times \cdots \times \dfrac{\sin{88^{\circ}}}{\cos{88^{\circ}}} \times \dfrac{\sin{89^{\circ}}}{\cos{89^{\circ}}} \right) }$

Note that $\cos{\theta} = \sin{\left( \dfrac{\pi}{2} - \theta \right)} \left( \color{#D61F06}{\text{Radians}} \right) \ \color{#333333}{}\implies \cos{x^{\circ}} = \sin{\left( 90 - x^{\circ} \right)} \left( \color{#3D99F6}{\text{Degrees}} \right)$

Hence, we have $\displaystyle \ln{\left(\tan{1^{\circ}} \times \tan{2^{\circ}} \times \cdots \times \tan{88^{\circ}} \times \tan{89^{\circ}}\right)} = \ \ln{ \left( \dfrac{\sin{1^{\circ}}}{\sin{89^{\circ}}} \times \dfrac{\sin{2^{\circ}}}{\sin{88^{\circ}}} \times \cdots \times \dfrac{\sin{88^{\circ}}}{\sin{2^{\circ}}} \times \dfrac{\sin{89^{\circ}}}{\sin{1^{\circ}}} \right) }$

Therefore, we see that $\ln{\displaystyle\left(\prod^{89}_{x=1}{\tan{x^{\circ}}}\right)} = \, \ln1 = \boxed{0}$