Question

Q). Show that: tan 75° + cot 75° = 4.​

Answer 1

Explanation:

`\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 \:  =  \: (tan45\degree + tan30\degree )/(1 - tan45\degree * tan30\degree )`

`\rm \:  =  \: (1 + (1)/( √(3) ) )/(1 - 1 * (1)/( √(3) ) )`

`\rm \:  =  \: (1 + (1)/( √(3) ) )/(1 - (1)/( √(3) ) )`

`\rm \:  =  \: ( √(3) + 1 )/( √(3) - 1)`

On rationalizing the denominator, we get

`\rm \:  =  \: ( √(3) + 1 )/( √(3) - 1) * ( √(3) + 1 )/( √(3) + 1 )`

`\rm \:  =  \: \frac{ {( √(3) + 1)}^(2) }{ {( √(3)) }^(2) - {(1)}^(2) }`

`\rm \:  =  \: (3 + 1 + 2 √(3) )/(3 - 1)`

`\rm \:  =  \: (4+ 2 √(3) )/(2)`

`\rm \:  =  \: (2(2+ √(3) ))/(2)`

`\rm \:  =  \: 2 + √(3)`

`\rm\implies \:\boxed{\tt{ tan75\degree = 2 + √(3) \: }}`

Now,

`\rm :\longmapsto\:cot75\degree`

`\rm \:  =  \: (1)/(tan75\degree )`

`\rm \:  =  \: (1)/(2 + √(3) )`

On rationalizing the denominator, we get

`\rm \:  =  \: (1)/(2 + √(3) ) * (2 - √(3) )/(2 - √(3) )`

`\rm \:  =  \: \frac{2 - √(3) }{ {(2)}^(2) - {( √(3)) }^(2) }`

`\rm \:  =  \: (2 - √(3) )/(4 - 3)`

`\rm \:  =  \: 2 - √(3)`

`\bf\implies \:\boxed{\tt{ cot75\degree = 2 - √(3) \: }}`

Now, Consider

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

`\rm \:  =  \: 2 + √(3) + 2 - √(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 + (1)/(tan75\degree )`

`\rm \:  =  \: \frac{ {tan}^(2)75\degree + 1}{tan75\degree }`

`\rm \:  =  \: \frac{1}{\frac{tan75\degree }{1 + {tan}^(2) 75\degree } }`

`\rm \:  =  \: \frac{2}{\frac{2tan75\degree }{1 + {tan}^(2) 75\degree } }`

We know,

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

`\rm \:  =  \: (2)/(sin150\degree )`

`\rm \:  =  \: (2)/(sin(180\degree - 30\degree ))`

`\rm \:  =  \: (2)/(sin30\degree )`

`\rm \:  =  \: 2 * 2`

`\rm \:  =  \: 4`

Hence,

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