Question

If $\cos x+\sin x=\frac{1}{2}$, Find the $\tan x$.

Answer 1

$\cos x+\sin x=1/2$

Divide both sides by $\cos x$

So, $1+\tan x=\frac{1}{2\cos x}=\frac{secx}{2}$

squaring both sides

$\Rightarrow {(1+\tan x)}^2=\frac{{\sec }^{2}x}{4}$

$\Rightarrow 4(1+{\tan }^{2}x+2\tan x)={\sec }^{2}x=(1+{\tan }^{2}x)$

$\Rightarrow 4(1+{\tan }^{2}x+2\tan x)=1+{\tan }^{2}x$

$\Rightarrow 3{\tan }^{2}x+8\tan x+3=0$

$\Rightarrow \tan x=\frac{-8\pm \sqrt{8^2-4\left(3\right)\left(3\right)}}{2.3}$

$=\frac{-8\pm \sqrt{64-36}}{6}$

$=\frac{-8\pm \sqrt{28}}{6}$

$=\frac{-8\pm 2\sqrt{7}}{6}$

$=\frac{-4\pm \sqrt{7}}{3}$

$\tan x=\frac{-4\pm \sqrt{7}}{3}$