Please prove that cos(A+B)cos(A-B)=cos^2A + cos^2B-1 ?

2 Answers
Shwetank Mauria
Feb 22, 2018

Please see below.

Explanation:

#cos(A+B)cos(A-B)#

= #(cosAcosB-sinAsinB)(cosAcosB+sinAsinB)#

= #cos^2Acos^2B-sin^2Asin^2B#

= #cos^2Acos^2B-(1-cos^2A)(1-cos^2B)#

= #cos^2Acos^2B-(1+cos^2Acos^2B-cos^2A-cos^2B)#

= #color(blue)(cos^2Acos^2B)-1color(red)(-cos^2Acos^2B)+cos^2A+cos^2B#

= #cos^2A + cos^2B-1#

P dilip_k
Feb 22, 2018

#LHS=cos(A+B)cos(A-B)#

#=(cosA cosB-sinAsinB)(cosA cosB+sinAsinB)#

#=(cos^2Acos^2B-sin^2Asin^2B)#

#=(cos^2Acos^2B-(1-cos^2A)(1-cos^2B)#

#=(cos^2Acos^2B-(1-cos^2A-cos^2B+cos^2Acos^2B)#

#=(cos^2Acos^2B-1+cos^2A+cos^2B-cos^2Acos^2B)#

#=cos^2A+cos^2B-1=RHS#

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