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.!