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]`