Trigonometry

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$${\text{co}}{{\text{s}}^2}\theta + {\text{co}}{{\text{s}}^4}\theta = 1$$
$$ ⇒ {\text{co}}{{\text{s}}^4}\theta = 1 - {\cos ^2}\theta $$
$$ ⇒ {\text{co}}{{\text{s}}^4}\theta = {\sin ^2}\theta $$
$$ ⇒ {\cos ^2}\theta .{\cos ^2}\theta = {\sin ^2}\theta $$
$$ ⇒ {\cos ^2}\theta = \frac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}$$
$$ ⇒ {\text{co}}{{\text{s}}^2}\theta = {\text{ta}}{{\text{n}}^2}\theta $$
$$ ⇒ {\text{ta}}{{\text{n}}^2}\theta + {\text{ta}}{{\text{n}}^4}\theta $$
$$ ⇒ {\text{co}}{{\text{s}}^2}\theta + {\text{co}}{{\text{s}}^4}\theta = 1$$

$${\text{If }}\tan \theta = 1$$
$${\text{It means }}\theta = {45^ \circ }$$
$$ = \frac{{8\sin \theta + 5\cos \theta }}{{{{\sin }^3}\theta - 2{{\cos }^3}\theta + 7\cos \theta }}$$
$$ = \frac{{8\sin {{45}^ \circ } + 5\cos {{45}^ \circ }}}{{{{\sin }^3}{{45}^ \circ } - 2{{\cos }^3}{{45}^ \circ } + 7\cos {{45}^ \circ }}}$$
$$ = \frac{{8 \times \frac{1}{{\sqrt 2 }} + 5 \times \frac{1}{{\sqrt 2 }}}}{{{{\left( {\frac{1}{{\sqrt 2 }}} \right)}^3} - 2{{\left( {\frac{1}{{\sqrt 2 }}} \right)}^3} + 7\left( {\frac{1}{{\sqrt 2 }}} \right)}}$$
$$ = 2$$

$${\cos ^2}\theta = \frac{{{{\left( {x + y} \right)}^2}}}{{4xy}}$$
$${\text{Max value of }}{\cos ^2}\theta = 1$$
$$ ⇒ 1 = \frac{{{{\left( {x + y} \right)}^2}}}{{4xy}}$$
$$ ⇒ 4xy = {\left( {x + y} \right)^2}$$
$$ ⇒ 4xy = {x^2} + {y^2} + 2xy$$
$$ ⇒ 0 = {x^2} + {y^2} - 2xy$$
$$ ⇒ 0 = {\left( {x - y} \right)^2}$$
$$ ⇒ 0 =×- y$$
$$ ⇒ x = y$$

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

cos1° cos2° cos3° ............. cos177° cos178° cos179°
= cos90°
= 0[0 will make whole series 0]
= 0