To achieve the desired form for each equation, we'll use various trigonometric identities:
A) cos^2(x) = 1 - sin^2(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
cos^2(x) = cos^2(x)
Both sides of the equation are already in the same form, so no further transformation is needed.
B) sin(x)cos^2(x) + sin^3(x) = sin(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
sin(x)cos^2(x) + sin^3(x) = sin(x)(1 - sin^2(x))
Next, using the distributive property, we can simplify further:
sin(x)cos^2(x) + sin^3(x) = sin(x) - sin^3(x)
Now, both sides of the equation are in the same form.
C) 1 - 2cos^2(x) = -1 + 2sin^2(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
1 - 2cos^2(x) = -1 + 2(1 - cos^2(x))
Simplifying further:
1 - 2cos^2(x) = -1 + 2 - 2cos^2(x)
Now, both sides of the equation are in the same form.
D) sin(-x)sin(x) = -1 + cos^2(x)
Using the identity sin(-x) = -sin(x), we can rewrite the equation as:
-sin(x)sin(x) = -1 + cos^2(x)
Simplifying further:
-sin^2(x) = -1 + cos^2(x)
Now, both sides of the equation are in the same form.
By using the appropriate trigonometric identities and simplifying the expressions, we have achieved the desired form for each equation.