Question

Show that tan 75 + cot 75 = 4.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:tan75\degree + cot75\degree$

Consider

$\rm :\longmapsto\:tan75\degree$

$\rm \:  =  \: tan(45\degree + 30\degree )$

$\rm \:  =  \: \dfrac{tan45\degree + tan30\degree }{1 - tan45\degree \times tan30\degree }$

$\rm \:  =  \: \dfrac{1 + \dfrac{1}{ \sqrt{3} } }{1 - 1 \times \dfrac{1}{ \sqrt{3} } }$

$\rm \:  =  \: \dfrac{1 + \dfrac{1}{ \sqrt{3} } }{1 - \dfrac{1}{ \sqrt{3} } }$

$\rm \:  =  \: \dfrac{ \sqrt{3} + 1 }{ \sqrt{3} - 1}$

On rationalizing the denominator, we get

$\rm \:  =  \: \dfrac{ \sqrt{3} + 1 }{ \sqrt{3} - 1} \times \dfrac{ \sqrt{3} + 1 }{ \sqrt{3} + 1 }$

$\rm \:  =  \: \dfrac{ {( \sqrt{3} + 1)}^{2} }{ {( \sqrt{3}) }^{2} - {(1)}^{2} }$

$\rm \:  =  \: \dfrac{3 + 1 + 2 \sqrt{3} }{3 - 1}$

$\rm \:  =  \: \dfrac{4+ 2 \sqrt{3} }{2}$

$\rm \:  =  \: \dfrac{2(2+ \sqrt{3} )}{2}$

$\rm \:  =  \: 2 + \sqrt{3}$

$\rm\implies \:\boxed{\tt{ tan75\degree = 2 + \sqrt{3} \: }}$

Now,

$\rm :\longmapsto\:cot75\degree$

$\rm \:  =  \: \dfrac{1}{tan75\degree }$

$\rm \:  =  \: \dfrac{1}{2 + \sqrt{3} }$

On rationalizing the denominator, we get

$\rm \:  =  \: \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} }$

$\rm \:  =  \: \dfrac{2 - \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3}) }^{2} }$

$\rm \:  =  \: \dfrac{2 - \sqrt{3} }{4 - 3}$

$\rm \:  =  \: 2 - \sqrt{3}$

$\bf\implies \:\boxed{\tt{ cot75\degree = 2 - \sqrt{3} \: }}$

Now, Consider

$\rm :\longmapsto\:tan75\degree + cot75\degree$

$\rm \:  =  \: 2 + \sqrt{3} + 2 - \sqrt{3}$

$\rm \:  =  \: 4$

Hence,

$\rm\implies \:\boxed{\tt{ tan75\degree + cot75\degree = 4 \: }}$

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Alternative Method

$\rm :\longmapsto\:tan75\degree + cot75\degree$

$\rm \:  =  \: tan75\degree + \dfrac{1}{tan75\degree }$

$\rm \:  =  \: \dfrac{ {tan}^{2}75\degree + 1}{tan75\degree }$

$\rm \:  =  \: \dfrac{1}{\dfrac{tan75\degree }{1 + {tan}^{2} 75\degree } }$

$\rm \:  =  \: \dfrac{2}{\dfrac{2tan75\degree }{1 + {tan}^{2} 75\degree } }$

We know,

$\rm :\longmapsto\:\boxed{\tt{ \frac{2tanx}{1 + {tan}^{2} x} = sin2x}}$

$\rm \:  =  \: \dfrac{2}{sin150\degree }$

$\rm \:  =  \: \dfrac{2}{sin(180\degree - 30\degree )}$

$\rm \:  =  \: \dfrac{2}{sin30\degree }$

$\rm \:  =  \: 2 \times 2$

$\rm \:  =  \: 4$

Hence,

$\rm\implies \:\boxed{\tt{ tan75\degree + cot75\degree = 4 \: }}$