How would you verify the identity?

sin^1/2x-cosx-sin^5/2xcosx=cos^3xradsinx

what is rad?

radical

multiply both sides by the sqrt sinx

sinx-cosxsinx-sin^3x cosx=cos^3x sinx
sinx-cosxsinx-(1-cos^2x)sinxcosx=
sinx(1-cosx)-sinxcosx+sinxcox^3x=
sinx(1-cosx-cosx)+sinx cox^3x=
Hmmm. This is telling me the identity does not exist, because sinx(1-2cosx) is not zero.

check my work. There has to be an error here, but I am certain this is (was ) leading somewhere.

I read your identity as

(sinx)^(1/2) - cosx - (sinx)^(5/2)*cosx = (cosx)^3 *(sinx)^(1/2)

I picked 30 degrees, and the
left side was NOT equal to the right side.

So the way you typed it, it is not an identity.

Thanks, Reiny.

I think I have the same book as you Crystal... I have no clue how to solve this thing... :p

To verify the identity sin^(1/2)x - cos(x) - sin^(5/2)x*cos(x) = cos^(3)x * rad(sin(x)), we need to manipulate one side of the equation to make it look like the other side.

Let's analyze the left side of the equation and simplify it step by step.

1. Start with the left side: sin^(1/2)x - cos(x) - sin^(5/2)x*cos(x).
2. Replace sin^(1/2)x with (sin(x))^(1/2) and sin^(5/2)x with (sin(x))^(5/2).
3. Apply the identity sin^2(x) + cos^2(x) = 1. Square both sides: (sin(x))^2 + (cos(x))^2 = 1.
4. Use the trigonometric Pythagorean identity: (sin(x))^2 = 1 - (cos(x))^2.
5. Substitute the expression from step 4 into the left side of the equation: (1 - (cos(x))^2)^(1/2) - cos(x) - ((1 - (cos(x))^2)^(5/2))*cos(x).
6. Apply the exponent rules: (1 - (cos(x))^2)^(1/2) = [1 - (cos(x))^2]^(1/2) = [(cos(x))^2 - 1]^(-1/2), and (1 - (cos(x))^2)^(5/2) = [(cos(x))^2 - 1]^(5/2).
7. Combine the above result with the left side of the equation: [(cos(x))^2 - 1]^(-1/2) - cos(x) - [(cos(x))^2 - 1]^(5/2)*cos(x).

Now let's simplify the right side of the equation: cos^(3)x * rad(sin(x)).

1. Start with the right side: cos^(3)x * rad(sin(x)).
2. Apply the identity (sin(x))^2 + (cos(x))^2 = 1. Solve for sin(x): sin(x) = (1 - (cos(x))^2)^(1/2).
3. Substitute this value into the right side of the equation: cos^(3)x * [(1 - (cos(x))^2)^(1/2)].
4. Since cos^2(x) = 1 - sin^2(x), we can rewrite the above expression: cos^(3)x * [(cos^2(x))^(1/2)].
5. Simplify the exponent: cos^(3)x * (cos(x))^(1/2).
6. Apply the power rule of exponents: cos^(3)x * cos^(1/2)x.
7. Combine the above result: cos^(7/2)x.

After simplifying both sides of the equation, we get:

[(cos(x))^2 - 1]^(-1/2) - cos(x) - [(cos(x))^2 - 1]^(5/2)*cos(x) = cos^(7/2)x.

Now you can compare the two sides of the equation to verify if they are identical.