Prove & Solve Formula ? 2cos(a+b)cos(a-b) = Cos2a+Cos2b

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Answer 1 · 2018-05-25T06:40:15

Please refer to Explanation.

The easiest way to prove this is by using the following Identity :

#2cosxcosy=cos(x+y)+cos(x-y).........(ast)#.

Let #x=a+b and y=a-b#.

#:. x+y=2a and x-y=2b#.

Utilising these in #(ast)#, we have,

#2cos(a+b)cos(a-b)=cos2a+cos2b#, as desired!

Answer 2 · 2018-05-25T06:58:43

Here is a Second Proof.

Here is a Second Proof using the Identity :

#cos2theta=2cos^2theta-1#.

Expanding #cos(a+b) and cos(a-b)#, we have,

#2cos(a+b)cos(a-b)#,

#=2(cosacosb-sinasinb)(cosacosb+sinasinb)#,

#=2{(cosacosb)^2-(sinasinb)^2}#,

#=2{cos^2acos^2b-sin^2asin^2b}#,

#=2{cos^2acos^2b-(1-cos^2a)(1-cos^2b)}#,

#=2{cos^2acos^2b-(1-cos^2a-cos^2b+cos^2acos^2b)}#,

#=2(cos^2a+cos^2b-1)#,

#=2cos^2a+2cos^2b-2#,

#=2cos^2a+2cos^2b-1-1#,

#=(2cos^2a-1)+(2cos^2b-1)#,

#=cos2a+cos2b#, as before!

Enjoy Maths.!