How do you verify the identity `(sin^3beta+cos^3beta)/(sinbeta+cosbeta)=1-sinbetacosbeta`?

There are 3 approaches to verifying the identity.

• Manipulate the left side into the form of the right side

• Manipulate the right side into the form of the left side

• Manipulate both sides until a point is reached where they are
`color(white)(x)"both equal"`

Using the first approach.

`"left side "=(sin^3beta+cos^3beta)/(sinbeta+cosbeta)`

The numerator is a `color(blue)"sum of cubes"` and can be factorised in general as follows.

`color(red)(bar(ul(|color(white)(2/2)color(black)(a^3+b^3=(a+b)(a^2+ab+b^2))color(white)(2/2)|)))`

`"here " a=sinbeta" and " b=cosbeta`

`rArrsin^3beta+cos^3beta=`

`=(sinbeta+cosbeta)(sin^2beta-sinbetacosbeta+cos^2beta)`

The left side can now be simplified.

`((cancel(sinbeta+cosbeta))(sin^2beta-sinbetacosbeta+cos^2beta))/(cancel(sinbeta+cosbeta))`

`=sin^2beta-sinbetacosbeta+cos^2beta`

`"Using the identity " color(red)(bar(ul(|color(white)(2/2)color(black)(sin^2beta+cos^2beta=1)color(white)(2/2)|)))`

`"Then" sin^2beta-sinbetacosbeta+cos^2beta=1-sinbetacosbeta`

`"Thus left side = right side "rArr"verified"`