Prove that - Tan 60-1 -tan 60-1 square -1-cos 30 - 1-cos 30

Hello LHS: tan 60 = √3 Thus we get [(√3 + 1)/( √3 – 1)]square Taking the conjugate(√3 + 1) and multiply numertor and denominator we get (28 + 16√3)/4 = 7 + 4√3 ...

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Standard XII

Mathematics

Local Maxima

Prove that : ...

Question

Prove that :
$((tan 60 + 1) \div (tan 60 - 1))^{2} = (1 + cos 30) \div (1 - cos 30)$

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Solution

LHS: $((tan 60 + 1) \div (tan 60 - 1))^{2} = (\frac{\sqrt{3} + 1}{\sqrt{3} - 1})^{2}$
Taking the conjugate (√3 + 1) and multiply numerator and denominator
$= ((\frac{\sqrt{3} + 1 \times}{\sqrt{3} - 1})^{2} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1})^{2}$
= $(\frac{(\sqrt{3} + 1)^{2}}{(\sqrt{3})^{2} - 1^{2}})^{2}$
$= (\frac{3 + 1 + 2 \sqrt{3}}{3 - 1})^{2}$
$= (\frac{4 + 2 \sqrt{3}}{2})^{2}$
$= (2 + \sqrt{3})^{2}$
$= 4 + 3 + 4 \sqrt{3}$
Thus, LHS $= 7 + 4 \sqrt{3}$
R.H.S $= (1 + cos 30) \div (1 - cos 30)$
$= \frac{1 + \frac{\sqrt{3}}{2}}{1 - \frac{\sqrt{3}}{2}}$
$= \frac{2 + \sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$
$= \frac{(2 + \sqrt{3})^{2}}{2^{2} - (\sqrt{3})^{2}}$
$= \frac{4 + 3 + 4 \sqrt{3}}{4 - 3}$
Thus, R.H.S $= 7 + 4 \sqrt{3}$
Therefore, LHS=RHS
Hence, $((tan 60 + 1) \div (tan 60 - 1))^{2} = (1 + cos 30) \div (1 - cos 30)$

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Similar questions

Show that:

(i) $\frac{1 - sin 60^{\circ}}{cos 60^{\circ}} = \frac{tan 60^{\circ} - 1}{tan 60^{\circ} + 1}$

(ii) $\frac{cos 30^{\circ} + sin 60^{\circ}}{1 + sin 30^{\circ} + cos 60^{\circ}} = cos 30^{\circ}$

Q. Show that:
(i)

\frac{1 - \sin 60 °}{\cos 60 °} = \frac{\tan 60 ° - 1}{\tan 60 ° + 1}

(ii)

\frac{\cos 30 ° + \sin 60 °}{1 + \sin 30 ° + \cos 60 °} = \cos 30 °

Q. Solve

\sqrt{\frac{1 + cos 30}{1 - cos 30}}

Q. Prove that :

\frac{c o s 30^{\circ} + s i n 60^{\circ}}{1 + c o s 60^{\circ} + s i n 30^{\circ}} = \frac{\sqrt{3}}{2}

Find the value of

$\frac{1 + s i n 30^{\circ}}{c o s 30^{\circ}} + \frac{c o s 30^{\circ}}{1 + s i n 30^{\circ}}$

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Prove that :((tan60+1)÷(tan60−1))2=(1+cos30)÷(1−cos30)

Prove that :
$((tan 60 + 1) \div (tan 60 - 1))^{2} = (1 + cos 30) \div (1 - cos 30)$