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Question Number 167594 by infinityaction last updated on 20/Mar/22

	Answered by som(math1967) last updated on 20/Mar/22
	$cosx+cosy+cosz=pcos(x+y+z) cosxcos(x+y+z)+cosycos(x+y+z) +coszcos(x+y+z)=pcos^2 (x+y+z) ........case i) same way sinxsin(x+y+z)+sinysin(x+y+z) +sinzsin(x+y+z)=psin^2 (x+y+z) .......case ii) casei)+caseii) cosxcos(x+y+z)+sinxsin(x+y+z) +cosycos(x+y+z)+sinysin(x+y+z) coszcos(x+y+z)+sinzsin(x+y+z) =p{sin^2 (x+y+z)+cos^2 (x+y+z)} cos(x+y+z−x)+cos(x+y+z−y) +cos(x+y+z−z)=p×1 ∴cos(x+y)+cos(y+z)+cos(z+x)=p$
	$c o s x + c o s y + c o s z = p c o s (x + y + z)$ $c o s x c o s (x + y + z) + c o s y c o s (x + y + z)$ $+ c o s z c o s (x + y + z) = {p c o s}^{2} (x + y + z)$ $. . . . . . . . c a s e i)$ $s a m e w a y$ $s i n x s i n (x + y + z) + s i n y s i n (x + y + z)$ $+ s i n z s i n (x + y + z) = {p s i n}^{2} (x + y + z)$ $. . . . . . . c a s e i i)$ $c a s e i) + c a s e i i)$ $c o s x c o s (x + y + z) + s i n x s i n (x + y + z)$ $+ c o s y c o s (x + y + z) + s i n y s i n (x + y + z)$ $c o s z c o s (x + y + z) + s i n z s i n (x + y + z)$ $= p {{s i n}^{2} (x + y + z) + {c o s}^{2} (x + y + z)}$ $c o s (x + y + z - x) + c o s (x + y + z - y)$ $+ c o s (x + y + z - z) = p \times 1$ $∴ c o s (x + y) + c o s (y + z) + c o s (z + x) = p$
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