Question

Evaluate:
$\frac{{\sec }^2{54}^o-{\cot }^2{36}^o}{co{\sec }^2{57}^o-{\tan }^2{33}^o}+2{\sin }^2{38}^o{\sec }^2{52}^o-{\sin }^2{45}^o+\frac{2}{\sqrt{3}}\tan {17}^o\tan {60}^o\tan {73}^o$

• A. $\frac{6}{5}$
• B. $\frac{7}{3}$
• C. $\frac{9}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{9}{2}$

$\frac{{\sec }^2{54}^o-{\cot }^2{36}^o}{co{\sec }^2{57}^o-{\tan }^2{33}^o}+2{\sin }^2{38}^o{\sec }^2{52}^o-{\sin }^2{45}^o+\frac{2}{\sqrt{3}}\tan {17}^o\tan {60}^o\tan {73}^o$

$\Rightarrow \frac{{\sec }^2{(90}^o-{36}^o)-{\cot }^2{36}^o}{co{\sec }^2{(90}^o-{33}^o)-{\tan }^2{33}^o}+2{\sin }^2{38}^o{\sec }^2{(90}^o-{38}^o)-{\sin }^2{45}^o+\frac{2}{\sqrt{3}}\tan ({90}^o-{73}^o)\tan {73}^o\tan {60}^o$

$\Rightarrow \frac{co{\sec }^2{36}^o-{\cot }^2{36}^o}{{\sec }^2{33}^o-{\tan }^2{33}^o}+2{\sin }^2{38}^{o}co{\sec }^2{38}^o-{\left(\frac{1}{\sqrt{2}}\right)}^2+\frac{2}{\sqrt{3}}\cot {73}^o\tan {73}^o\times \sqrt{3}$

$\Rightarrow \frac{1}{1}+2{\sin }^2{38}^o\times \frac{1}{{\sin }^2{38}^o}-\frac{1}{2}+\frac{2}{\sqrt{3}}\times \frac{1}{\tan {73}^o}\times \tan {73}^o\times \sqrt{3}$

$[\because co{\sec }^{2}0-{\cot }^{2}0=1,{\sec }^{2}0-{\tan }^{2}0=1]$

$\Rightarrow 1+2-\frac{1}{2}+2=5-\frac{1}{2}$

$\Rightarrow \frac{9}{2}$