Question

Question #431bc

Answer 1

`-sinx`

Explanation:

`lim_(delta->0)(cos(x+delta)-cos(x))/delta=-sinx`

I can't prove it.

Answer 2

See explanation

Explanation:

There are multiple ways to show this

A common way is to use the definition of a derivative,
which is nicely done here on the site

Another way is to use series expansions

Cosine and sine can expressed as the series expansions

• `cos(x)=1/(0!)-x^2/(2!)+x^4/(4!)-x^6/(6!)...`

• `sin(x)=x/(1!)-x^3/(3!)+x^5/(5!)-x^7/(7!)...`

Take the derivative of cosine

`d/dxcos(x)=d/dx(1/(0!)-x^2/(2!)+x^4/(4!)-x^6/(6!)...)`

`=-2x/(2!)+4x^3/(4!)-6x^5/(6!)...`

`=-x/(1!)+x^3/(3!)-x^5/(5!)...`

`=-(x/(1!)-x^3/(3!)+x^5/(5!)...)`

`=-sin(x)`