Question

Evaluate 5 sin^2 30° + cos^2 45°+4 tan^2 30° / 2 sin 30°cos 30° - tan 45°​

Answer 1

To evaluate:-

$\bigstar \sf \: \dfrac{5 \: {sin}^{2} 30 \degree + {cos}^{2} 45 \degree + 4 \: {tan}^{2} 30 \degree}{2 \: {sin}^{2}30 \degree \: cos \: 30 \degree - tan \: 45 \degree}$

Solution:-

$\bigstar \sf \: \dfrac{5 \: {sin}^{2} 30 \degree + {cos}^{2} 45 \degree + 4 \: {tan}^{2} 30 \degree}{2 \: {sin}^{2}30 \degree \: cos \: 30 \degree - tan \: 45 \degree}$

sin 30° = 1/2

cos 45° = 1/√2

tan 30° = 1/√3

cos 30° = √3/2

tan 45° = 1

Put all the values of ratio's,

$\implies \sf \dfrac{5 \times {( \dfrac{1}{2} )}^{2} + {( \dfrac{1}{ \sqrt{2} } )}^{2} + 4 \times {( \dfrac{1}{ \sqrt{3} } )}^{2} }{2 \times \dfrac{1}{2} \times \dfrac{ \sqrt{3} }{2} - 1}$

$\implies \sf \dfrac{5 \times \dfrac{5}{4} + \dfrac{1}{2} + 4 \times \dfrac{1}{3} }{2 \times \dfrac{1}{2} \times \dfrac{ \sqrt{3} }{2} - 1 }$

$\implies \sf \: \dfrac{ \dfrac{5}{4} + \dfrac{1}{2} + \dfrac{4}{3} }{ \dfrac{ \sqrt{3} }{2} - 1}$

$\implies \sf \: \dfrac{ \dfrac{15 + 6 + 16}{12} }{ \dfrac{ \sqrt{3} - 2}{2} }$

$\implies \sf \: \dfrac{37}{12} \times \dfrac{2}{ \sqrt{3} - 2}$

$\implies \sf \dfrac{5}{6( \sqrt{3} - 2) }$

Therefore,

$\bigstar \sf \: \dfrac{5 \: {sin}^{2} 30 \degree + {cos}^{2} 45 \degree + 4 \: {tan}^{2} 30 \degree}{2 \: {sin}^{2}30 \degree \: cos \: 30 \degree - tan \: 45 \degree} = \dfrac{5}{6( \sqrt{3} - 2)}$