Question

​$\boxed {\boxed{ { \red{ \bold{ \underline{Question:-}}}}}}$
Find the value of:-
$\frac{cos \: 60° + \: sin \: 45° - \: cot \: 30 °}{tan \: 60 ° \: + \: sec \: 45° - \: cosec \: 30° \: }$
​$\boxed {\boxed{ { \red{ \bold{ \underline{Question:-}}}}}}$
Evaluate:-
$8 \sqrt{3} \times \: {cosec}^{2} \: 30° \times \sin(60°) \times \cos(60°) \times {cos}^{2} 45° \times \sin(45°) \times \tan(30°) \times {cosec}^{3} 45°$
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Answer 1

Since,

cos 60° = 1/2

sin 45° = 1/√2

cot 30° = √3

tan 60° = √3

sec 45° = √2

cosec 30° = 2

Then,

$\frac{cos {60}^{0} + sin {45}^{0} - cot {30}^{0} }{tan {60}^{0} + sec {45}^{0} - cosec {30}^{0} } = \frac{ \frac{1}{2} + \frac{1}{ \sqrt{2} } - \sqrt{3} }{ \sqrt{3} + \sqrt{2} - 2 } \\ \\ = \frac{ \frac{ \sqrt{2} + 2 - 2 \sqrt{6} }{2 \sqrt{2} } }{ \sqrt{3 } + \sqrt{2} - 2} = \frac{ \sqrt{2} + 2 - 2 \sqrt{6} }{2 \sqrt{6} + 4 - 4 \sqrt{2} }$

Again,

$8 \sqrt{3} \times {cosec}^{2} {30}^{0} \times sin {60}^{0} \times cos {60}^{0} \times {cos}^{2} {45}^{0} \times sin {45}^{0} \times tan {30}^{0} \times {cosec}^{3} {45}^{0} \\ = 8 \sqrt{3} \times {2}^{2} \times \frac{ \sqrt{3} }{2} \times \frac{1}{2} \times {( \frac{1}{ \sqrt{2} }) }^{2} \times \frac{1}{ \sqrt{2} } \times \frac{1}{ \sqrt{3} } \times { \sqrt{2} }^{3} \\ = 8 \sqrt{3} \times 4 \times \frac{ \sqrt{3} }{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{ \sqrt{2} } \times \frac{1}{ \sqrt{3} } \times 2 \sqrt{2} \\ = 8 \sqrt{3 } \times 1 = 8 \sqrt{3}$

Hope It Helps :)

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