Solution:
$$\eqalign{ \frac{{3\sin {{58}^ \circ }}}{{\cos {{32}^ \circ }}} + \frac{{3\sin {{42}^ \circ }}}{{\cos {{48}^ \circ }}} \cr = \frac{{3\cos {{32}^ \circ }}}{{\cos {{32}^ \circ }}} + \frac{{3\cos {{48}^ \circ }}}{{\cos {{48}^ \circ }}} \cr \left\{ {\therefore \,{{58}^ \circ } + {{32}^ \circ } = {{90}^ \circ }\,\ \,{{42}^ \circ } + {{48}^ \circ } = {{90}^ \circ }} \right\} \cr = 3 + 3 \cr = 6 \cr} $$

Solution:
$$\eqalign{ \frac{{\cot {{30}^ \circ } - \cot {{75}^ \circ }}}{{\tan {{15}^ \circ } - \tan {{60}^ \circ }}} \cr = \frac{{\tan {{60}^ \circ } - \tan {{15}^ \circ }}}{{\tan {{15}^ \circ } - \tan {{60}^ \circ }}} \cr = \frac{{ - \left( {\tan {{15}^ \circ } - \tan {{60}^ \circ }} \right)}}{{\tan {{15}^ \circ } - \tan {{60}^ \circ }}} \cr = - 1 \cr} $$

Solution:
$$\eqalign{ r\sin \theta = 1 \cr r\cos \theta = \sqrt 3 \cr \Rightarrow \frac{{{\text{ }}rsin\theta }}{{{\text{ }}r\cos \theta }} = \frac{1}{{\sqrt 3 }} \cr \Rightarrow {\text{tan}}\theta = \frac{1}{{\sqrt 3 }} \cr \Rightarrow \sqrt 3 {\text{tan}}\theta = 1 \cr \left( {{\text{Add 1 both sides}}} \right) \cr \Rightarrow \sqrt 3 {\text{tan}}\theta + 1 = 1 + 1 \cr \Rightarrow \sqrt 3 {\text{tan}}\theta + 1 = 2 \cr} $$

Solution:
$$\eqalign{ \frac{{{{\cos }^4}\theta - {{\sin }^4}\theta + 2{{\sin }^2}\theta + 3}}{{\left( {{\text{cosec}}\,\theta + \cot \theta + 1} \right)\left( {{\text{cosec}}\,\theta - \cot \theta + 1} \right) - 2}} \cr = \frac{{\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)\left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) + 2{{\sin }^2}\theta + 3}}{{\left( {{\text{cosec}}\,\theta + 1 + \cot \theta } \right)\left( {{\text{cosec}}\,\theta + 1 - \cot \theta } \right) - 2}} \cr = \frac{{\left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) + 2{{\sin }^2}\theta + 3}}{{{{\left( {{\text{cosec}}\,\theta + 1} \right)}^2} - {{\cot }^2}\theta - 2}} \cr = \frac{{{{\cos }^2}\theta + {{\sin }^2}\theta + 3}}{{{\text{cose}}{{\text{c}}^2}\theta + 1 + 2{\text{cosec}}\,\theta - {{\cot }^2}\theta - 2}} \cr = \frac{4}{{2{\text{cosec}}\,\theta }} \cr = 2\sin \theta \cr} $$

Solution:
$$\eqalign{ \frac{{{\text{sin A}}}}{{1 + \cos {\text{ A}}}}{\text{ + }}\frac{{{\text{sin A}}}}{{1 - \cos {\text{ A}}}} \cr \Rightarrow \frac{{{\text{sin A}}\left( {1 - \cos {\text{ A}}} \right) + {\text{sin A}}\left( {1 + \cos {\text{ A}}} \right)}}{{\left( {1 + \cos {\text{ A}}} \right)\left( {1 - \cos {\text{ A}}} \right)}} \cr \Rightarrow \frac{{{\text{sin A}} - {\text{sin A}}{\text{.cosA}} + {\text{sin A}} + {\text{sin A}}{\text{.cosA}}}}{{{\text{1}} - {\text{co}}{{\text{s}}^2}{\text{A}}}} \cr \Rightarrow \frac{{2{\text{sin A}}}}{{{{\sin }^2}{\text{A}}}} \cr \Rightarrow 2{\text{ cosec A}} \cr} $$

Solution:
$$\eqalign{ {\left( {\frac{{{\text{sin 4}}{{\text{7}}^ \circ }}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\frac{{\cos {{43}^ \circ }}}{{{\text{sin }}{{47}^ \circ }}}} \right)^2} - 4{\text{co}}{{\text{s}}^2}{45^ \circ } \cr = {\left( {\frac{{{\text{cos 4}}{{\text{3}}^ \circ }}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\frac{{sin{{47}^ \circ }}}{{{\text{sin }}{{47}^ \circ }}}} \right)^2} - 4 \times \frac{1}{2} \cr = 1 + 1 - 2 \cr = 0 \cr {\bf{Note:}} \cr \left( {\sin \left( {{{90}^ \circ } - \theta } \right)} \right) = \cos \theta \cr} $$

Solution:
sinθ + cosθ = √5cosθ
sinθ = cosθ(√5 - 1)
$$\cot \theta = \frac{1}{{\sqrt 5 - 1}} \times \frac{{\sqrt 5 + 1}}{{\sqrt 5 - 1}} = \frac{{\sqrt 5 + 1}}{4}$$