Question

sqrt{1 + cos 30} \div \sqrt{1 - cos 30} - tan60 = cosec30
1+cos30
​
÷
1−cos30
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−tan60=cosec30
Prove ​
THE ANSWER OF THIS QUESTION ..​

Answer 1

Correct question :-

$\sf \small\sqrt{\dfrac{1 + cos30\degree}{1-cos30\degree}} = ?$

Solution :-

By rationalising the denominator ,

$\sf\dashrightarrow \small \sqrt{\dfrac{1 + cos30\degree}{1-cos30\degree}\times \dfrac{1+cos30\degree}{1+cos30\degree}}$

$\sf \dashrightarrow \small \sqrt{\dfrac{(1+cos30\degree)^2}{(1-cos30\degree)(1+cos30\degree)}}$

$\qquad \small\sf \bullet \; (a+b)(a-b) = a^2-b^2$

$\sf \dashrightarrow \small \dfrac{\sqrt{(1+cos30\degree)^2}}{\sqrt{1 - cos^230\degree}}$

$\sf \dashrightarrow \small \dfrac{1+cos30\degree}{\sqrt{sin^230\degree}}$

$\sf \dashrightarrow \small \dfrac{1}{sin30\degree} + \dfrac{cos30\degree}{sin30\degree}$

$\sf \dashrightarrow \small cosec30\degree + cot30\degree$

$\sf \dashrightarrow \small 2 + \sqrt3$

$\boldsymbol\therefore \underline{\sf \green{Hence , \; \sqrt{\dfrac{1 + cos30\degree}{1-cos30\degree}} = 2 + \sqrt3 }}$

Answer 2

$\sf \small\sqrt{\dfrac{1 + cos30\degree}{1-cos30\degree}} = ?$

Solution :-

• By rationalising the denominator ,

$\sf\dashrightarrow \small \sqrt{\dfrac{1 + cos30\degree}{1-cos30\degree}\times \dfrac{1+cos30\degree}{1+cos30\degree}}$

$\sf \dashrightarrow \small \sqrt{\dfrac{(1+cos30\degree)^2}{(1-cos30\degree)(1+cos30\degree)}}$

$\qquad \small\sf \bullet \; (a+b)(a-b) = a^2-b^2∙(a+b)$

$\sf \dashrightarrow \small \dfrac{\sqrt{(1+cos30\degree)^2}}{\sqrt{1 - cos^230\degree}}$

$\sf \dashrightarrow \small \dfrac{1+cos30\degree}{\sqrt{sin^230\degree}}$

$\sf \dashrightarrow \small \dfrac{1}{sin30\degree} + \dfrac{cos30\degree}{sin30\degree}$

$\sf \dashrightarrow \small cosec30\degree + cot30\degree$

$\sf \dashrightarrow \small 2 + \sqrt3$

$\boldsymbol\therefore \underline{\sf {Hence , \; \sqrt{\dfrac{1 + cos30\degree}{1-cos30\degree}} = 2 + \sqrt3 }}$