Final answer:
To expand the expression in powers of sin x, we use trigonometric identities to rewrite it in terms of sin x. After making the necessary substitutions and simplifying, the constant term in the expansion is 4. Therefore, the negative of the constant term is -4.
Step-by-step explanation:
To expand the given expression in powers of sin x, we need to use trigonometric identities to rewrite the expression in terms of sin x. Let's start by using the identity cos 2x = 1 - 2sin^2 x to replace the first term. Then, we can use the identity cos 4x = 2cos^2 2x - 1 to replace the fourth and fifth terms. After making these substitutions, the expression becomes:
1 - 2sin^2 x + sin^2 x + (2cos^2 2x - 1)sin^2 x + cos 2x + cos^2 x + cos 2x
Now, we can simplify the expression by combining like terms:
3cos 2x + 3sin^2 x + 2cos^2 2xsin^2 x + cos^2 x
The constant term in the expansion is obtained by setting x = 0. When we do this, we get:
3cos 0 + 3sin^2 0 + 2cos^2 0sin^2 0 + cos^2 0 = 3 + 0 + 0 + 1 = 4.
Therefore, the negative of the constant term in the expansion is -4.