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if A+B+C = pi/2, prove that cos^2A+cos^2B+cos^2C = 2+2 sinA sinB sinC

if A+B+C = pi/2, prove that cos^2A+cos^2B+cos^2C = 2+2 sinA sinB sinC

Grade:11

1 Answers

Pawan joshi
26 Points
5 years ago
=cos^2A+cos^2B+cos^2C
=1-sin^2A+cos^2B+cos^2C
=1+cos(A+B)cos(B-A)+cos^2C 
 ( cos^2B-sin^2A=cos(B+A)cos(B-A) )
=1+sinCcos(B-A)+cos^2C
=1+sinCcosAcosB +sinAsinBsinC+1-sin^2C
=2+sinC(cosAcosB-sinC)+sinAsinBsinC
=2+sinC(cosAcosB-cos(A+B))+sinAsinBsinC
=2+sinC(sinAsinB)+sinAsinBsinC
=2+2sinAsinBsinC.......
Hope this helps..
 
 

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