verify:
sin^3 x - cos^3 x = (sinx+cosx)(1-sinxcosx)
sin³x-cos³x
=(sin(x)-cos(x))(sin²x+sin(x)cos(x)+cos²(x))
=(sin(x)-cos(x))(1+sin(x)cos(x))
Either there is a typo in the question, or the identity is not valid.
To verify the given equation, we will simplify the left-hand side (LHS) and right-hand side (RHS) separately and then compare them.
Let's start with the LHS:
LHS: sin^3x - cos^3x
We can use the identity for the difference of cubes, which states that:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In our case, a = sinx and b = cosx. Applying the identity, we have:
sin^3x - cos^3x = (sinx - cosx)(sin^2x + sinx*cosx + cos^2x)
Now let's simplify the RHS:
RHS: (sinx + cosx)(1 - sinxcosx)
We can use the identity for the product of sums, which states that:
(a + b)(c + d) = ac + ad + bc + bd
Applying this identity to the RHS, we have:
(sin x + cos x)(1 - sin x cos x) = sin x - sin^2 x cos x + cos^2 x sin x - sin x cos x
Now, we need to simplify both sides of the equation and see if they are equal:
LHS = (sinx - cosx)(sin^2x + sinx*cosx + cos^2x)
Expanding the equation, we have:
LHS = sin^3x + sin^2x*cosx + sinx*cos^2x - cos^3x - sinx*cos^2x - sin^2x*cosx
Combining like terms, we get:
LHS = sin^3x - cos^3x
We can see that the LHS is indeed equal to the expression given.
Therefore, the verification is complete, and we have shown that sin^3x - cos^3x = (sinx + cosx)(1 - sinxcosx).
To verify the expression sin^3 x - cos^3 x = (sinx+cosx)(1-sinxcosx), we can use the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2) if we let a = sinx and b = cosx.
Using this identity, we have:
sin^3 x - cos^3 x = (sinx - cosx)(sin^2 x + sinx cosx + cos^2 x)
Now, we need to simplify sin^2 x + sinx cosx + cos^2 x.
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can substitute this into the expression:
sin^2 x + sinx cosx + cos^2 x = 1 + sinx cosx
Now, substituting this result back into the previous expression, we have:
(sin x - cos x)(1 + sin x cos x)
This is not the same as the expression (sinx+cosx)(1-sinxcosx). Therefore, the expression sin^3 x - cos^3 x is not equal to (sinx+cosx)(1-sinxcosx).