Question

Представьте в виде суммы
1) cos(π/3 - α) × cos α
2) cos(π/4 + α) × cos α

Answer 1

$Cos( frac{ pi }{3}- alpha ) *Cos alpha= frac{Cos( frac{ pi }{3} - alpha - alpha )+Cos( frac{ pi }{3}- alpha + alpha ) }{2}= frac{1}{2}[Cos( frac{ pi }{3}-2 alpha )+$$+Cos frac{ pi }{3}]$$= frac{1}{2} Cos( frac{ pi }{3} -2 alpha )- frac{1}{2}* frac{1}{2}= frac{1}{2}Cos( frac{ pi }{3}-2 alpha )- frac{1}{4}$

$Cos( frac{ pi }{4} + alpha )*Cos alpha = frac{Cos( frac{ pi }{4} + alpha - alpha )+Cos( frac{ pi }{4}+ alpha + alpha }{2}= frac{1}{2}[Cos frac{ pi }{4}+Cos( frac{ pi }{4} +$$+2 alpha )]= frac{1}{2} * frac{ sqrt{2} }{2}+ frac{1}{2}Cos( frac{ pi }{4} +2 alpha )= frac{ sqrt{2} }{4} + frac{1}{2} Cos( frac{ pi }{4}+2 alpha )$

Answer 2

1) (1/2)cos( π/3 - 2α) +1/4 * * * ⇔ (1/2)cos( 2α -π/3) +1/4 * * *

Answer 3

task/28692272
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Представьте в виде суммы 
1) cos(π/3 - α) * cos α
2) cos(π/4 + α) * cos α
------------  cosα*cosβ = (cos(α-β) +cos(α+β) ) /2 -----------
но , по другому 
1)
cos(π/3 - α) * cos α =( cos(π/3)*cosα +sin(π/3)*sinα )*cosα = 
= (1/2)*cos²α +(√3 /2) *sinα *cosα = (1/4)*(1+cos2α +√3 *sin2α )  = 
 1/4 + (1/2)*( (1/2)*cos2α +(√3/2) *sin2α ) =1/4 + (1/2)*cos(2α - π/3) .
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2) 
cos(π/4 + α) * cos α =(cos(π/4) * cosα -  sin(π/4) * sinα)*cos α =
(1/√2)*( cos²α - sinα*cos α ) =(1/2√2)*( 1+cos2α - sin2α ) =
1/2√2  +(1/2)* ( (1/√2)*cos2α - (1/√2)*sin2α ) = √2 / 4  +(1/2)*cos(2α +π/4) .