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Answer 1

Step-by-step explanation:

To evaluate :-

$\sf \star \: \dfrac {tan^{2} \: 60\degree + 4 \: sin ^{2} \: 45 \degree + 3 \: sec ^{2} \: 30\degree + 5 \: cos ^{2} \: 90\degree}{cosec \: 30\degree + sec \: 60\degree - cot^{3} \: 30\degree}$

Solution :-

We have,

$\sf \star \: \dfrac {tan^{2} \: 60\degree + 4 \: sin ^{2} \: 45 \degree + 3 \: sec ^{2} \: 30\degree + 5 \: cos ^{2} \: 90\degree}{cosec \: 30\degree + sec \: 60\degree - cot^{3} \: 30\degree}$

tan 60° = √3

sin 45° = 1/√2

sec 30° = 2/√3

cos 90° = 0

cosec 30° = 2

sec 60° = 2

cot 30° = √3

Put all values :

$\sf \longrightarrow \dfrac{( \sqrt{3} ) ^{2} + 4 \times \bigg(\dfrac{1}{ \sqrt{2}} \bigg)^{2} + 3 \times { \bigg( \dfrac{2}{ \sqrt{3} } \bigg)}^{2} + 5 \times (0)^{2} }{ 2 + 2 - {( \sqrt{3}) }^{2} }$

$\sf \longrightarrow \dfrac{3 + 4 \times \dfrac{1}{2} + 3 \times \dfrac{4}{3} + 5 \times 0 }{4 - 3}$

$\sf \longrightarrow \dfrac{3 + 2 + 4 + 0}{1}$

$\longrightarrow \boxed{\bold{9}}$

Therefore,

$\bold{ \star \: \dfrac {tan^{2} \: 60\degree + 4 \: sin ^{2} \: 45 \degree + 3 \: sec ^{2} \: 30\degree + 5 \: cos ^{2} \: 90\degree}{cosec \: 30\degree + sec \: 60\degree - cot^{3} \: 30\degree} = 9}$