<s>[INST] Determine value of $m$ Consider $m = \tan x + \frac{\sin x + \cos x}{\sin x - \cos x}$ . Now determine $m$ so that expression has solution .
I used Wolfram to find range of $f(x) = \tan x + \frac{\sin x + \cos x}{\sin x - \cos x}$ but it was unable to find it!
 [/INST] $m = \frac{\sin x}{\cos x} + \frac{\sin x + \cos x}{\sin x - \cos x}$
$m = \frac{\sin x(\sin x - \cos x) +\cos x(\sin x + \cos z)}{\cos x(\sin x - \cos x)}$
$m = \frac{\sin^{2}x + \cos^{2}x}{\sin x \cos x - \cos^{2}x}$
$(\sin x \cos x - \cos^{2}x)m - 1 = 0$
$(\tan x - 1)m - (1+\tan^{2}x) = 0$
$tan^{2}x - m\tan x +m+1 = 0$
This has real solutions when the discriminant is greater than or equal to $0$:
$m^{2} - 4(m+1) \geq 0 \Leftrightarrow (m+2)^{2} \geq 0$
So it has solutions $\forall m \in \mathbb{R}$
The range is $\tan x \in \mathbb{R}$
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