We have to prove that [(cos x)^3 - (sin x)^3]/[cos x - sin x]= 1 + cos x * sin x

Starting with the left hand side:

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

use a^3 - b^3 = (a - b)(a^2 + ab + b^2)

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We have to prove that [(cos x)^3 - (sin x)^3]/[cos x - sin x]= 1 + cos x * sin x

Starting with the left hand side:

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

use a^3 - b^3 = (a - b)(a^2 + ab + b^2)

=> [(cos x - sin x)[(cos x)^2 + cos x * sin x + (sin x)^2]]/( cos x - sin x)

cancel (cos x - sin x)

=> [(cos x)^2 + cos x * sin x + (sin x)^2]

Use (cos x)^2 + (sin x)^2 = 1

=> 1 + cos x * sin x

This is the right hand side.

**This proves that [(cos x)^3 - (sin x)^3]/[cos x - sin x] = 1 + cos x * sin x**