Write the expression in terms of costheta and then simplify.

cos^4theta - sin^4theta + sin^2theta

Ans: cos^4 θ - 1 - cos^4 θ + 1 - cos^2 θ

= -cos^2 θ

cos^4 Ø - sin^4 Ø + sin^2 Ø

= (cos^2 Ø + sin^2 Ø)(cos^2 Ø - sin^2 Ø) + sin^2 Ø
= (1)(cos^2 Ø - sin^2 Ø) + sin^2 Ø
= cos^2 Ø

To simplify the expression in terms of cosθ, we'll use the trigonometric identity:

sin^2θ = 1 - cos^2θ

First, let's rewrite the expression by substituting sin^2θ with its corresponding identity:

cos^4θ - (1 - cos^2θ)^2 + sin^2θ

Expanding (1 - cos^2θ)^2, we get:

cos^4θ - (1 - 2cos^2θ + cos^4θ) + sin^2θ

Now, simplify the expression:

cos^4θ - 1 + 2cos^2θ - cos^4θ + sin^2θ

Rearranging the terms:

cos^4θ - cos^4θ + 2cos^2θ + sin^2θ - 1

Simplify further:

2cos^2θ + sin^2θ - 1

Since the trigonometric identity sin^2θ + cos^2θ = 1, we have:

2cos^2θ + (1 - cos^2θ) - 1

Combine like terms:

2cos^2θ + 1 - cos^2θ - 1

Finally, simplify:

cos^2θ

Therefore, the simplified expression in terms of cosθ is cos^2θ.