Question

Cos (pi + x) + cos (pi-x) ?

Answer 1

`color (blue)(-2cos x)`

Explanation:

We have
`Cos (pi + x) + cos (pi-x)`

Identities that we are going to use are as follows
`color (red)(cos (a+b)=cos a cos b-sin a sin b)`
And

`color (red)(cos (a-b)=cos a cos b+sin a sin b)`

It is quite clear that
`cos (a-b)+cos (a+b)=2cos a cos b`

Plugging in values for a and b, we get
`Cos (pi + x) + cos (pi-x)=2cos pi cos x=-2cos x` {Note:`cos pi=-1`}