Solving problem with compound angle formula

Find the value of
$\displaystyle (\ \text{i}\ )\ \ \ \ \frac{{\sin {{{80}}^{\circ }}}}{{\sin {{{20}}^{\circ }}}}-\frac{{\sqrt{3}}}{{2\sin {{{80}}^{\circ }}}}$

$\displaystyle (\ \text{ii}\ )\ \ \ \ \left( {\cot {{{25}}^{\circ }}-1} \right)\left( {\cot {{{24}}^{\circ }}-1} \right)\left( {\cot {{{20}}^{\circ }}-1} \right)\left( {\cot {{{21}}^{\circ }}-1} \right)$

Solution

$\displaystyle (\ \text{i}\ )\ \ \ \ \frac{{\sin {{{80}}^{\circ }}}}{{\sin {{{20}}^{\circ }}}}-\frac{{\sqrt{3}}}{{2\sin {{{80}}^{\circ }}}}$

$\displaystyle \ \ \ \ \ \ =\frac{{\cos {{{10}}^{\circ }}}}{{2\sin {{{10}}^{\circ }}\cos {{{10}}^{\circ }}}}-\frac{{\sqrt{3}}}{{2\cos {{{10}}^{\circ }}}}$ 

$\displaystyle \ \ \ \ \ \ =\frac{1}{{2\sin {{{10}}^{\circ }}}}-\frac{{\sqrt{3}}}{{2\cos {{{10}}^{\circ }}}}$ 

$\displaystyle \ \ \ \ \ \ =\frac{{\frac{1}{2}}}{{\sin {{{10}}^{\circ }}}}-\frac{{\frac{{\sqrt{3}}}{2}}}{{\cos {{{10}}^{\circ }}}}$

$\displaystyle \ \ \ \ \ \ =\frac{{\sin {{{30}}^{\circ }}}}{{\sin {{{10}}^{\circ }}}}-\frac{{\cos {{{30}}^{\circ }}}}{{\cos {{{10}}^{\circ }}}}$

$\displaystyle \ \ \ \ \ \ =\frac{{\sin {{{30}}^{\circ }}\cos {{{10}}^{\circ }}-\cos {{{30}}^{\circ }}\sin {{{10}}^{\circ }}}}{{\sin {{{10}}^{\circ }}\cos {{{10}}^{\circ }}}}$ 

$\displaystyle \ \ \ \ \ \ =\frac{{\sin \left( {{{{30}}^{\circ }}-{{{10}}^{\circ }}} \right)}}{{\sin {{{10}}^{\circ }}\cos {{{10}}^{\circ }}}}$

$\displaystyle \ \ \ \ \ \ =\frac{{\sin {{{20}}^{\circ }}}}{{\sin {{{10}}^{\circ }}\cos {{{10}}^{\circ }}}}$

$\displaystyle \ \ \ \ \ \ =\frac{{2\sin {{{10}}^{\circ }}\cos {{{10}}^{\circ }}}}{{\sin {{{10}}^{\circ }}\cos {{{10}}^{\circ }}}}$ 

$\displaystyle \ \ \ \ \ \ =2$ 

$\displaystyle (\ \text{ii}\ )\ \ \ \ \left( {\cot {{{25}}^{\circ }}-1} \right)\left( {\cot {{{24}}^{\circ }}-1} \right)\left( {\cot {{{20}}^{\circ }}-1} \right)\left( {\cot {{{21}}^{\circ }}-1} \right)$ 

$\displaystyle \ \ \ \ \ \ \ =\left( {\frac{1}{{\tan {{{25}}^{\circ }}}}-1} \right)\left( {\frac{1}{{\tan {{{24}}^{\circ }}}}-1} \right)\left( {\frac{1}{{\tan {{{20}}^{\circ }}}}-1} \right)\left( {\frac{1}{{\tan {{{21}}^{\circ }}}}-1} \right)$

$\displaystyle \ \ \ \ \ \ \ =\left( {\frac{{1-\tan {{{25}}^{\circ }}}}{{\tan {{{25}}^{\circ }}}}} \right)\left( {\frac{{1-\tan {{{24}}^{\circ }}}}{{\tan {{{24}}^{\circ }}}}} \right)\left( {\frac{{1-\tan {{{20}}^{\circ }}}}{{\tan {{{20}}^{\circ }}}}} \right)\left( {\frac{{1-\tan {{{21}}^{\circ }}}}{{\tan {{{21}}^{\circ }}}}} \right)$

$\displaystyle \ \ \ \ \ \ \ =\left( {\frac{{\left( {1-\tan {{{25}}^{\circ }}} \right)\left( {1-\tan {{{20}}^{\circ }}} \right)}}{{\tan {{{25}}^{\circ }}\tan {{{20}}^{\circ }}}}} \right)\left( {\frac{{\left( {1-\tan {{{24}}^{\circ }}} \right)\left( {1-\tan {{{21}}^{\circ }}} \right)}}{{\tan {{{24}}^{\circ }}\tan {{{21}}^{\circ }}}}} \right)$

$\displaystyle \ \ \ \ \ \ \ =\left( {\frac{{1-\tan {{{20}}^{\circ }}-\tan {{{25}}^{\circ }}+\tan {{{25}}^{\circ }}\tan {{{20}}^{\circ }}}}{{\tan {{{25}}^{\circ }}\tan {{{20}}^{\circ }}}}} \right)\left( {\frac{{1-\tan {{{21}}^{\circ }}-\tan {{{24}}^{\circ }}+\tan {{{24}}^{\circ }}\tan {{{21}}^{\circ }}}}{{\tan {{{24}}^{\circ }}\tan {{{21}}^{\circ }}}}} \right)$ 

$\displaystyle \ \ \ \ \ \ \ =\left( {\frac{{2\tan {{{25}}^{\circ }}\tan {{{20}}^{\circ }}}}{{\tan {{{25}}^{\circ }}\tan {{{20}}^{\circ }}}}} \right)\left( {\frac{{2\tan {{{24}}^{\circ }}\tan {{{21}}^{\circ }}}}{{\tan {{{24}}^{\circ }}\tan {{{21}}^{\circ }}}}} \right)$

$\displaystyle \ \ \ \ \ \ \ =4$ 

$\displaystyle *Note$

$\displaystyle \tan {{45}^{\circ }}=\tan \left( {{{{25}}^{\circ }}+{{{20}}^{\circ }}} \right)=\frac{{\tan {{{25}}^{\circ }}+\tan {{{20}}^{\circ }}}}{{1-\tan {{{25}}^{\circ }}\tan {{{20}}^{\circ }}}}$ 

$\displaystyle \tan {{45}^{\circ }}=\tan \left( {{{{24}}^{\circ }}+{{{21}}^{\circ }}} \right)=\frac{{\tan {{{24}}^{\circ }}+\tan {{{21}}^{\circ }}}}{{1-\tan {{{24}}^{\circ }}\tan {{{21}}^{\circ }}}}$

$\displaystyle \tan {{45}^{\circ }}=1$ 

$\displaystyle \therefore \ \ \tan {{25}^{\circ }}\tan {{20}^{\circ }}=1-\tan {{25}^{\circ }}-\tan {{20}^{\circ }}\ \ and\ \ \tan {{24}^{\circ }}\tan {{21}^{\circ }}=1-\tan {{24}^{\circ }}-\tan {{21}^{\circ }}\ $