Question

(cos a)^2 - (cos a)^4 + (sin a)^4

Answer 1

$\cos^2\alpha-\cos^4\alpha+\sin^4\alpha=\cos^2\alpha-(\cos^4\alpha-\sin^4\alpha)=\cos^2 \alpha-\\ \\ -(\cos^2\alpha-\sin^2\alpha)(\cos^2\alpha+\sin^2\alpha)=\cos^2\alpha-\cos^2\alpha+\sin^2\alpha=\sin^2\alpha$

Answer 2

To simplify the expression, let's use the identity:

1 - (cos a)^2 = (sin a)^2

Substituting this identity into the expression, we have:

(cos a)^2 - (cos a)^4 + (sin a)^4 = (1 - (sin a)^2) - (cos a)^4 + (sin a)^4 = 1 - (sin a)^2 - (cos a)^4 + (sin a)^4

Now, let's combine the terms that have a common factor of (sin a)^2:

= 1 - (sin a)^2 + (sin a)^4 - (cos a)^4

Notice that (sin a)^4 - (cos a)^4 can be factored as the difference of squares:

= 1 - (sin a)^2 + [(sin a)^2 - (cos a)^2][(sin a)^2 + (cos a)^2] = 1 - (sin a)^2 + (sin a)^2 - (cos a)^2

Since (sin a)^2 + (cos a)^2 = 1, we can simplify further:

= 1 - (sin a)^2 + (sin a)^2 - (1 - (sin a)^2) = 1 - (sin a)^2 + (sin a)^2 - 1 + (sin a)^2 = (sin a)^2

Therefore, the simplified expression is (sin a)^2.