Question

Evaluate

tan260°+4 sin245°+ 3 sec 30° + 5 cos290°
_______________________
cosec 30° + sec 60°- cot230°​

Answer 1

Correct question :

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

Given :

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

To find :

• Evaluate

Solution :

$\implies \sf{\dfrac {tan^{2} 60 \degree + sin^{2} 45 \degree + 3 \: sec^{2} 30 \degree + 5 \: cos^{2} 90 \degree}{cosec \: 30 \degree + sec \: 60 \degree - cot^{2}30 \degree }} \\ \\ \\ \implies \sf \dfrac{ (\sqrt{3})^{2} + 4 \times (\frac{1}{ \sqrt{2} })^{2} + 3 \times (\frac{2}{ \sqrt{3} })^{2} + 5 \times 0^{2}}{2 + 2 - (\sqrt{3})^{2} } \\ \\ \\ \implies \sf{ (\dfrac{3 + 4 \times \frac{1}{2} + 3 \frac{4}{3} + 5 \times 0 }{4 - 3})} \\ \\ \\ \implies \sf{ \dfrac{3 + 2 + 4 + 0}{1} } \\ \\ \\ \implies{\boxed{\sf{ \red {\underline{9}}}}}$

Answer 2

Answer:

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

⠀

$\bf{Evaluate :}$

$:\implies \sf\dfrac {tan^{2} 60 \degree + sin^{2} 45 \degree + 3 \: sec^{2} 30 \degree + 5 \: cos^{2} 90 \degree}{cosec \: 30 \degree + sec \: 60 \degree - cot^{2}30 \degree } \\\\\\:\implies \sf \dfrac{ \bigg(\sqrt{3}\bigg)^{2} + 4 \times \bigg(\dfrac{1}{ \sqrt{2} }\bigg)^{2} + 3 \times \bigg(\dfrac{2}{ \sqrt{3} }\bigg)^{2} + 5 \times \bigg(0\bigg)^{2}}{2 + 2 - \bigg(\sqrt{3}\bigg)^{2} } \\\\\\ :\implies \sf{\dfrac{3 + 4 \times {}^{1}\!/{}_{2} + 3 \times {}^{4}\!/{}_{3} + 5 \times 0 }{4 - 3}} \\\\\\:\implies \sf{ \dfrac{3 + 2 + 4 + 0}{1} } \\\\\\:\implies\underline{\boxed{\sf{9}}}$

⠀

$\therefore\:\underline{\textsf{Hence, the required answer is \textbf{9}}}.$

$\rule{180}{1.5}$

$\bigstar\:\sf Trigonometric\:Values :\\\begin{tabular}{|c|c|c|c|c|c|}\cline{1-6}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}& Not D \hat{e} fined \\\cline{1-6}\end{tabular}$