Question

Please explain the answer. Deepest explanation will be marked as the brainliest. Please don't post irrelevant answers.​

Answer 1

Step-by-step explanation:

Let

$x = \alpha + \beta ....(i) \: \: and \: \: y = \alpha - \beta ...(ii)$

Now, adding and subtracting (i) and (ii), we get,

$\alpha = \frac{x + y}{2} ....(iii) \: \: and \: \: \beta = \frac{x - y}{2}.....(iv)$

Now, we have,

$\cos(x) + \cos(y) = \cos( \alpha + \beta ) + \cos( \alpha - \beta ) .......(from \: \: (i) \: \: and \: \: (ii))$

$= \cos( \alpha ) \cos( \beta ) - \sin( \alpha ) \sin( \beta ) + \cos( \alpha ) \cos( \beta ) + \sin( \alpha ) \sin( \beta )$

$= 2 \cos( \alpha ) \cos( \beta)$

$= 2 \cos( \frac{x + y}{2} ) \cos( \frac{x - y}{2} ) ....from \: \: (iii) \: \: and \: \: (iv)$

Hence, proved