Question

See the attached picture OR

Show that tan 75º =(√3 +1) /(√3-1)= 2 + √3

Hence deduce that tan 75º - cot 75º = 4 sin 60°.​




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Answer 1

$\bold{ \huge{ \mathfrak{Hello!}}}$

$\bold{ \tan(75) - \cot(75) = 4 \sin(60) }$

$\longrightarrow \: \tan(75) - \frac{1}{ \tan(75)} = 2 \sqrt{3}$

$\longrightarrow { \tan }^{2} (75) - 2 \sqrt{3} \tan(75) - 1 = 0$

$\longrightarrow \tan(75) = \frac{ 2 \sqrt{3} ± \sqrt{ {( - 2 \sqrt{3} )}^{2} - 4 \times 1 \times ( - 1) }}{2 \times 1}$

$\longrightarrow \: \tan(75) = \frac{2 \sqrt{3} ± \sqrt{16} }{2}$

$\longrightarrow \: \tan(75) = \sqrt{3} ± 2$

$\bold{as( \sqrt{3 } - 2) < 0 \: thats \: why \: its \: not \: acceptable }$

Hance,

$\longrightarrow \: \boxed {\mathtt{\tan(75) = \sqrt{3} + 2}} \: \: \bold{Proved}$