Question

This can be achieved by transforming one or both sides of the given equation,

until they look the same.
A) cos^2(x) = 1-sin^2(x)B) sin(x)cos^2+sin^3(x) = sin(x)
c) 1-2cos^2(x) = -1+2sin^2(x)
d) sin(-x)sin(x) = -1+cos^2(x)

Answer 1

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.