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.