<s>[INST] Find $\tan x $ if $\sin x+\cos x=\frac12$ It is given that $0 < x < 180^\circ$ and $\sin x+\cos x=\frac12$, Find $\tan x $. 
I tried all identities I know but I have no idea how to proceed.  Any help would be appreciated. 
 [/INST] Here is a geometric way of looking at this problem. I will change the angle variable to $t.$ We need to find $\tan t$ if $\sin t + \cos t = 1/2.$ I will use $x = \cos t, y = \sin t$ so  we need to solve $$x^2 + y^2 = 1, x + y = 1/2$$ for $x, y.$ that is, to find the two points where the line cuts the unit circle. By eliminating one variable, say $x,$ gives 
$$1 = x^2 +(1/2-x)^2= 2x^2 - 1/2x + 1/4$$ which is a quadratic equation $$8x^2 - 4x -3 = 0 $$ in $x.$  Now use the quadratic formulae to find $x,$ then $y = 1/2 -x$ and $\tan t = y/x ={1\over 2x } - 1$ can then be computed.  Solving the quadratic equation for 
$${1 \over x} = {-4 \pm 2\sqrt{28} \over 6}  = {2(-1 \pm \sqrt 7)/3},\ 
{1 \over 2x} -1= {-4 \pm 2\sqrt{28} \over 6}  = {(-4 \pm \sqrt 7) \over 3}.
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