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Question Number 129360 by bramlexs22 last updated on 15/Jan/21

 { ((sin a + sin b = (1/3))),((cos a+cos b = (5/3))) :}      cos^2  (a+b) =?

{sina+sinb=13cosa+cosb=53cos2(a+b)=?

Commented by bramlexs22 last updated on 15/Jan/21

 nice trigonometry

nicetrigonometry

Answered by liberty last updated on 15/Jan/21

 (1) cos (a−b)= (4/9)  (2) sin (a+b)+sin (a+b)cos (a−b)=(5/9)  ⇒ sin (a+b)+(4/9)sin (a+b)=(5/9)  ⇒((13)/9)sin (a+b)=(5/9) ⇒sin (a+b)=(5/(13))  ⇒ cos^2 (a+b)=1−((25)/(169))=((144)/(169))

(1)cos(ab)=49(2)sin(a+b)+sin(a+b)cos(ab)=59sin(a+b)+49sin(a+b)=59139sin(a+b)=59sin(a+b)=513cos2(a+b)=125169=144169

Answered by bramlexs22 last updated on 15/Jan/21

⇔ ((2sin (((a+b)/2))cos (((a−b)/2)))/(2cos (((a+b)/2))cos (((a−b)/2)))) = (1/5)  ⇔ tan (((a+b)/2))=(1/5)  ⇒ tan (a+b) = ((2tan (((a+b)/2)))/(1−tan^2 (((a+b)/2))))  ⇒ tan (a+b) = ((2/5)/(1−(1/(25)))) =((10)/(24))=(5/(12))  ⇒sec^2 (a+b)=1+((25)/(144))=((169)/(144))  ⇒ cos^2 (a+b)=((144)/(169))

2sin(a+b2)cos(ab2)2cos(a+b2)cos(ab2)=15tan(a+b2)=15tan(a+b)=2tan(a+b2)1tan2(a+b2)tan(a+b)=251125=1024=512sec2(a+b)=1+25144=169144cos2(a+b)=144169

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