For α∈R, cosαcosx+siny≥sinx, ∀x∈R, then find the sum of the possible values of sinα+siny.


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Question Number 27958 by bmind4860 last updated on 17/Jan/18

$F o r α \in R, c o s α c o s x + s i n y ⩾ s i n x, \forall x \in R,$ $t h e n f i n d t h e s u m o f t h e p o s s i b l e v a l u e s$ $o f s i n α + s i n y .$
	Answered by ajfour last updated on 17/Jan/18

	$\sin y ⩾ \sin x - \cos α \cos x$ $\Rightarrow ⩾ \sqrt{1 + \cos^{2} α} [\frac{1}{\sqrt{1 + \cos^{2} α}} \sin x - \frac{\cos α}{\sqrt{1 + \cos^{2} α}} \cos x]$ $\sin y ⩾ \sqrt{1 + \cos^{2} α} \sin [x - \tan^{- 1} (\cos α)]$ $s i n c e x c a n b e a n y v a l u e,$ $a n d \sin y ⩽ 1 \Rightarrow \cos α = 0$ $\Rightarrow \sin α = \pm 1$ $t h e n t h i s i m p l i e s$ $\sin y ⩾ \sin x$ $a n d a g a i n f o r t h i s t o b e a l w a y s$ $t r u e \sin y = 1$ $h e n c e \sin y + \sin α = 0 o r 2 .$
	Commented by bmind4860 last updated on 17/Jan/18

	$t h a n k y o u! w o u l d y o u s u g g e s t w h a t d o$ $y o u t h i n k a b o u t t h e f o l l o w i n g q u e s t i o n,$ $w h e t h e r i t i s r i g h t o r w r o n g$ $f i n d g e n e r a l s o l u t i o n, s i n x + s i n 3 x + s i n \sqrt{x} = 0 .$
	Commented by ajfour last updated on 17/Jan/18

	$c a n t s a y, i {c o u l d n}^{'} t f i n d a w a y .$
	Commented by bmind4860 last updated on 17/Jan/18

	$m e t o o {t h a t}^{'} s w h y i t h i n k i t i s w r o n g$ $a n d t h a n k y o u f o r s u g g e s t i o n$
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