Simplify the expression (sin^3 A - cos^3 A) / (sin A - cos A), where A is an acute angle.

Question

Accepted Answer

Given expression: (sin^3 A - cos^3 A) / (sin A - cos A)

We use the algebraic identity for difference of cubes:

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

Here, x = sin A and y = cos A.

So, sin^3 A - cos^3 A = (sin A - cos A)(sin^2 A + sin A cos A + cos^2 A)

Substitute back into the expression:

(sin^3 A - cos^3 A) / (sin A - cos A) = (sin A - cos A) 
(sin^2 A + sin A cos A + cos^2 A) / (sin A - cos A)

The (sin A - cos A) terms cancel out:

= sin^2 A + sin A cos A + cos^2 A

Using the Pythagorean identity:

sin^2 A + cos^2 A = 1

Therefore:

= 1 + sin A cos A

Hence, the simplified expression is 1 + sin A cos A.

Answer

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