Question

sin30°+tan45°-cosec60° / sec30°+cos60°+cot45°
​

Answer 1

$\purple \bullet \footnotesize \tt{ \: sin 30° \: = \frac{1}{2} }$

$\purple \bullet \footnotesize \tt{ \: tan45° \: = 1 }$

$\purple \bullet \footnotesize \tt{ \: cosec60°\: = \frac{2}{ \sqrt{3} } }$

$\purple \bullet \footnotesize \tt{ \: sec30°\: = \frac{2}{ \sqrt{3} } }$

$\purple \bullet \footnotesize \tt{ \: cos 60 °\: = \frac{1}{2} }$

$\purple \bullet \footnotesize \tt{ \: cot45° \: = 1 }$

$\: \: \: \: \: \: \: \: \: \: \: \: \:$

$\bullet \footnotesize \tt \purple{ \: Solution:- }$

$\: \: \: \: \: \: \: \: \: \: \: \: \:$

$\purple\leadsto\tt{ \frac{sin30° \: + \: tan45 ° \: - \: cosec60°}{sec30 ° \: + \: cos60° \: + \: cot45°} }$

$\purple\leadsto\tt{ \frac{ \frac{1}{2} \: + \: 1 \: - \: \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } \: + \: \frac{1}{2} \: + \: 1} }$

$\purple\leadsto\tt{ \frac{ \frac{ \sqrt{3 } \: + \: 2 \sqrt{3} \: - \: 4}{ \sqrt{3} } }{ \frac{4 \: + \: \sqrt{3} \: + \: 2 \sqrt{3} }{ \sqrt{3} } } }$

$\purple\leadsto\tt{ \frac{ \frac{ 3 \sqrt{3} \: - \: 4}{ \sqrt{3} } }{ \frac{3 \sqrt{3} \: + \: 4}{ \sqrt{3} } } }$

$\purple\leadsto \tt{ \frac{3 \sqrt{3} \: - \: 4}{ \sqrt{3} } \times \frac{ \sqrt{3} }{3 \sqrt{3} \: + \: 4 } }$

$\purple\leadsto\tt{ \frac{3 \sqrt{3} \: - \: 4}{3 \sqrt{3} \: + \: 4 } }$

$\purple\leadsto\tt{ \frac{3 \sqrt{3} \: - \: 4}{3 \sqrt{3} \: + \: 4 } } \times \frac{3 \sqrt{3} \: - \: 4}{3 \sqrt{3} \: - \: 4}$

$\purple\leadsto\tt{ \frac{(3 \sqrt{3} \: - \: 4) ^{2} }{(3 \sqrt{3})^{2} \: - \: (4 )^{2} } }$

$\purple\leadsto\tt{ \frac{(3 \sqrt{3}) ^{2} \: - \: 2 \times 3 \sqrt{3} \times 4 \: + \: (4) ^{2} }{27 \: - \: 16 } }$

$\purple\leadsto\tt{ \frac{27 \: - \: 24 \sqrt{3} \: + \: 16}{27 \: - \: 16 } }$

$\footnotesize\boxed{\boxed{\purple\leadsto\tt{ \frac{43 \: - \: 24 \sqrt{3} }{11 } }}}$

Answer 2

$\Huge{\texttt{{{\color{Magenta}{⛄A}}{\red{N}}{\purple{S}}{\pink{W}}{\blue{E}}{\green{R}}{\red{♡}}{\purple{࿐⛄}}{\color{pink}{:}}}}}$

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✠We know that:-

$\tt \purple\longrightarrow{Sin30° = \frac{1}{2} }$

$\tt \purple \longrightarrow{Tan45° = 1}$

$\tt \purple \longrightarrow{Cosec60° = \frac{2}{ \sqrt{3} } }$

$\tt \purple \longrightarrow{Sec30° = \frac{2}{ \sqrt{3} }}$

$\tt \purple \longrightarrow{Cos60° = \frac{1}{2} }$

$\tt \purple \longrightarrow{Cot45° = 1}$

✠Given us,

$\boxed{ \tt{ \frac{Sin30° + Tan45° - Cosec60°}{Sec30° + Cos60° + Cot45°}}}$

$\tt \orange{ \leadsto \frac{1}{2} + \frac{ \frac{1 - 2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } } + \frac{1}{2} + 1 }$

$\tt \orange{ \leadsto \frac{3}{2} - \frac{ \frac{ 2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } } + \frac{3}{2} }$

$\tt \orange {\leadsto 3 \sqrt{3} - \frac{4}{4} + 3 \sqrt{3} }$

$\tt \orange{ \leadsto3 \sqrt{3} - \frac{4}{4} + {3 \sqrt{3} \times 3\sqrt{3} - \frac{4}{3 \sqrt{3} } - 4 }}$

$\tt \orange{ \leadsto 27 + 16 - \frac{24 \sqrt{3}}{27} - 16}$

$\huge \boxed{ \tt { 43 - \frac{24 \sqrt{3} }{11}}}$

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$\huge\red{\boxed{\orange{\mathcal{{{\fcolorbox{red}{i}{{\red{@Master}}}}}}}}}$