[tex]\boxed {\boxed{ { \red{ \bold{ \underline{Question:-}}}}}}[/tex]
Find the value of:-
[tex] \frac{cos \: 60° + \: sin \: 45° - \: cot \: 30 °}{tan \: 60 ° \: + \: sec \: 45° - \: cosec \: 30° \: } [/tex]
[tex]\boxed {\boxed{ { \red{ \bold{ \underline{Question:-}}}}}}[/tex]
Evaluate:-
[tex]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°[/tex]
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Verified answer
Since,
cos 60° = 1/2
sin 45° = 1/√2
cot 30° = √3
tan 60° = √3
sec 45° = √2
cosec 30° = 2
Then,
[tex] \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} } [/tex]
Again,
[tex]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} [/tex]
Hope It Helps :)
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