Question

cot²30°-2sec²45°+3tan²60°-2/5cosec³30°​

Answer 1

To evaluate:-

$\bigstar \sf \: \dfrac{ {cot}^{3} \: 30 \degree - 2 \: {sec}^{2} \: 45 \degree + 3 \: {tan}^{2} \: 60 \degree - 2}{5 \: {cosec}^{3} \: 30 \degree}$

Solution:-

$\bigstar \sf \: \dfrac{ {cot}^{3} \: 30 \degree - 2 \: {sec}^{2} \: 45 \degree + 3 \: {tan}^{2} \: 60 \degree - 2}{5 \: {cosec}^{3} \: 30 \degree}$

cot 30° = √3

sec 45° = √2

tan 60° = √3

cosec 30° = 2

Put all values,

$\implies \sf \dfrac{ { (\sqrt{3} )}^{3} - 2 \times {( \sqrt{2} )}^{2} + 3 \times { (\sqrt{3}) }^{2} - 2}{5 \times {(2)}^{3} }$

$\implies \sf \dfrac{3 \sqrt{3} - 4 + 9 \sqrt{3} - 2 }{5 \times 8}$

$\implies \sf \dfrac{3 \sqrt{3} + 9 \sqrt{3} - 6}{40}$

$\implies \sf \dfrac{ \cancel{12} \times \sqrt{6} - 6}{ \cancel{40}}$

$\implies \sf \dfrac{3 \sqrt{6} - 6}{10}$

Therefore,

$\sf \: \dfrac{ {cot}^{3} \: 30 \degree - 2 \: {sec}^{2} \: 45 \degree + 3 \: {tan}^{2} \: 60 \degree - 2}{5 \: {cosec}^{3} \: 30 \degree} = \dfrac{3 \sqrt{6} - 6 }{10}$