Simplify this Cosine Problem

Question

sin^2x-cos x (cos(-x)) sin^2x

Answer 1

Since cosx is an even function, cos(-x) = cosx.

So, sin²x - cosx(cos(-x))sin²x = sin²x(1 - cos²x) = sin²x sin²x = sin⁴x

Answer 2

cos(x) = cos(-x)

See below for an explanation of this identity*

1st substitute cos(x) where we see cos(-x), so we now have:

sin²x-(cos(x)cos(x)sin²x)

This simplifies to:

sin²x-cos²xsin²x

Since the GCF of the expression is sin²x, we can factor that out:

sin²x (1-cos²x)

Using the identity sin²x+cos²x=1, we know that 1-cos²x=sin²x. So we can substitute that identity and get:

sin²x (sin²x)

Using laws of exponents, we can add the exponents and get:

sin⁴x.

*The function f(x)=cosx is symmetric across the y-axis. That means that any x-value that we plug in will have the same y-value as any (-x) value we plug in.

for example:

If we plug in (π/3) for x, we get (1/2) for y. If we plug in (-π/3) for x, again we get (1/2) for y (since (-π/3) is in quadrant 4 and has the same reference angle as π/3 and cosine is positive in Q4).

This will happen for all x-values. So we get a new identity that cos(x) = cos(-x).