Question

If $2{\sin }^{2}x+{\cos }^2{45}^{\circ }=\tan {45}^{\circ }$ and x is an acute angle, then the value of $\tan x$ is

• A. 1
• B. $\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. 3

Answer 1

Answer: (C)

$\frac{1}{\sqrt{3}}$

$2{\sin }^{2}x+{\cos }^2{45}^{\circ }=\tan {45}^{\circ }\Rightarrow 2{\sin }^{2}x+{\left(\frac{1}{\sqrt{2}}\right)}^2=1\Rightarrow 2{\sin }^{2}x=1\frac{1}{2}=\frac{1}{2}\Rightarrow {\sin }^{2}x=\frac{1}{4}\Rightarrow \sin x=\frac{1}{2}=\sin {30}^{\circ }\Rightarrow x=30{}^{\circ }\therefore \tan x=\tan {30}^{\circ }=\frac{1}{\sqrt{3}}$