Question

Evaluate:
$\:\sf\dfrac {tan^{2} 60 \degree + sin^{2} 45 \degree + 3 \: sec^{2} 30 \degree + 5 \: cos^{2} 90 \degree}{cosec \: 30 \degree + sec \: 60 \degree - cot^{2}30\degree }$
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Answer 1

${\underline{\underline{\frak{To\;Evaluate \;:}}}}$

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

⠀⠀

${\underline{\underline{\frak{Solution\;:}}}}$

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

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

$:\implies\sf {\dfrac{3 + 4 \times {}^{1}\!/{}_{2} + 3 \times {}^{4}\!/{}_{3} + 5 \times 0 }{4 - 3}}$

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

$:\implies\sf 9$

$\qquad\qquad\qquad{\underline{\underline{\sf{\pink{So,\;Done!}}}}}$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━

$\qquad\qquad\qquad\boxed{\underline{\underline{\purple{\bigstar \: \bf\: More\:to\:know\:\bigstar}}}}$

$\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3} }{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }& 1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}$

Answer 2

${\underline{\underline{\frak{To\;Evaluate \;:}}}}$

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

⠀⠀

${\underline{\underline{\frak{Solution\;:}}}}$

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

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

$:\implies\sf {\dfrac{3 + 4 \times {}^{1}\!/{}_{2} + 3 \times {}^{4}\!/{}_{3} + 5 \times 0 }{4 - 3}}$

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

$:\implies\sf 9$

$\qquad\qquad\qquad{\underline{\underline{\sf{\orange{So,\;Done!}}}}}$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━

$\qquad\qquad\qquad\boxed{\underline{\underline{\green{\bigstar \: \bf\: More\:to\:know\:\bigstar}}}}$

$\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3} }{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }& 1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}$