Question

(cos A + cos b)^2 + (sin A + sin B)^2=?

Answer 1

First, we expand the equation:

$cos^{2}A+2cosAcosB+ cos^{2}B+sin^{2}A+2sinAsinB+sin^{2}B$

Then, we combine certain terms in order to simplify them using trigonometry identities.

$cos^{2}A+sin^{2}A+cos^{2}B+sin^{2}B+2cosAcosB+2sinAsinB$

Note these identities:
Pythagoran relation: $cos^{2}A+sin^{2}A = cos^{2}B+sin^{2}B=1$
Sum and Difference Formula/Identities: $cos(A-B) = cosAcosB+sinAsinB$

Thus, if we apply these identities, the simplified equation would be:
$2(cos(A-B))+2$

Simplying further, the answer would be

$2[cos(A-B)+1]$