Verify the Identity:
(1-cosx)(1+cosx)=sin^2x
(1-cosx)(1+cosx)=sin^2x
1 - cos^2 x = sin^2 x
but sin^2x + cos^2 x = 1 identity
sin^2 x = sin^2 x
To verify the given identity (1-cosx)(1+cosx) = sin^2x, we need to simplify both sides of the equation and demonstrate that they are equivalent.
Let's start with the left side of the equation:
(1 - cosx)(1 + cosx)
To simplify this expression, we can use the distributive property of multiplication:
1 * 1 + 1 * cosx - cosx * 1 - cosx * cosx
Simplifying further, we have:
1 + cosx - cosx - cos^2x
Notice that the -cosx and +cosx terms cancel out:
1 - cos^2x
Now, let's rewrite sin^2x using the Pythagorean identity: sin^2x = 1 - cos^2x.
Substituting this into the equation, we get:
1 - cos^2x = sin^2x
This verifies the given identity.
To summarize, we simplified both sides of the equation and showed that they were equivalent by using the distributive property and the Pythagorean identity sin^2x = 1 - cos^2x.