<s>[INST] Proving a trigonometric inequality $(1-\sin a)x^2 -2x\cos a + 1+ \sin a \ge 0$ $(1-\sin (a))x^2 -2x\cos(a) + 1+ \sin( a) \ge 0$, where $a,x$ are any two real constants.
Any suggestions on how to prove this ? I tried playing with it, but nothing useful came out.
 [/INST] $\cos^2 a = 1-\sin^2a=(1+\sin a)(1-\sin a)$
$1+\sin a = \frac {\cos^2 a}{1-\sin a}$
Substituting this value in the equation -- 
$(1-\sin a)x^2 -2x\cos(a) + [1+ \sin( a)]  $ 
=$(1-\sin a)x^2 -2x\cos(a) + \frac {\cos^2 a}{1-\sin a}  $
=$[x \root  \of {1-\sin a} - \frac {\cos a}{\root \of {1-\sin a}}  ]^2$
$\ge 0$
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