Question

How do you prove `(1+cosx)(1-cosx)=sin^2x`?

Answer 1

`sin^2x + cos^2x = 1`

Explanation:

the identity known is `sin^2x + cos^2x = 1`.

this can be rearranged to give `1 - cos^2x = sin^2x`.

using the 'difference of two squares' identity,

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

`(1+cosx)(1-cosx) = 1^2 - cos^2x`

`1^2 = 1`

`(1+cosx)(1-cosx) = 1 - cos^2x`

since `1 - cos^2x = sin^2x`, `(1+cosx)(1-cosx) = sin^2x`.

Answer 2

FOIL the LHS and use the trig identity that `sin^2x+cos^2x=1`

Explanation:

First, FOIL the left hand side (LHS):

`(1+cosx)(1-cosx)=1cancel(-cosx+cosx)+cos^2x`

`=1-cos^2x`

Now, re-write the equation:

`1-cos^2x=sin^2x`

There is a trig identity that states:

`sin^2x+cos^2x=1`

Substitute the 1 in our proof:

`sin^2xcancel(+cos^2x-cos^2x)=sin^2x`

`sin^2x=sin^2x`