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Answer:

Alright, let's prove the identity you mentioned using some trigonometric identities. Here's the step-by-step process:

Start with the left side of the equation:

(sin^3 x + cos^3 x)/(sin x + cos x)

Now, let's factor the numerator:

(sin x + cos x)(sin^2 x - sin x * cos x + cos^2 x)/(sin x + cos x)

Next, we can cancel out the common factor of (sin x + cos x):

(sin^2 x - sin x * cos x + cos^2 x)

Now, let's simplify the numerator using the identity sin^2 x + cos^2 x = 1:

(1 - sin x * cos x - sin x * cos x + cos^2 x)

Combine like terms:

(1 - 2sin x * cos x + cos^2 x)

Finally, using the identity cos^2 x = 1 - sin^2 x, we can rewrite the expression as:

(1 - 2sin x * cos x + 1 - sin^2 x)

Simplify further:

(2 - 2sin x * cos x - sin^2 x)

And voila! We have arrived at the right side of the equation: 1 - sin x * cos x.

Therefore, we have successfully proven the identity: (sin^3 x + cos^3 x)/(sin x + cos x) = 1 - sin x * cos x.