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

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) .