Question

How do you rewrite (sin x - cos x)(sin x + cos x)? Thank you.

Answer 1

`sin^2x-cos^2x`

Explanation:

You're probably used to dealing with this only in quadratics, but the expression is in the difference of squares pattern

`(a-b)(a+b)=a^2-b^2`

where `color(red)(a=sinx)` and `color(blue)(b=cosx)`

We can just plug in our values for `a` and `b` into our difference of squares expression, and we get:

`color(red)((sinx))^2-color(blue)((cosx))^2`

Which can be rewritten as

`sin^2x-cos^2x`

If you don't believe me, we can FOIL this expression to make sure:

With FOIL, we multiply the first, outside, inside and last terms and add the result. Thus, we have:

• First terms: `sinx*sinx=color(red)(sinx^2)`
• Outside terms: `sinx*cosx=sinxcosx`
• Inside terms: `sinx*-cosx=-sinxcosx`
• Last terms: `-cosx*cosx=-color(blue)(cosx^2)`

Now we have

`color(red)((sinx))^2+cancel(sinxcosx-sinxcosx)-color(blue)((cosx))^2`

The middle terms obviously cancel out, and we can rewrite this as

`sin^2x-cos^2x`

The key realization is that the original expression in question was in a difference of squares pattern.

If you have a `(a-b)(a+b)` pattern, you can always use the difference of squares identity to quickly simplify it.

Hope this helps!

Answer 2

`- cos 2x`

Explanation:

`f(x) = (sin x - cos x)(sin x + cos x) = (sin^2 x - cos^2 x)`
Reminder of trig identity:
`cos 2x = cos^2 x - sin^2 x`
Finally,
`f(x) = - cos 2x`