Question

I need to establish the identity.
(sin^3theta+cos^3theta)/(sin theta + cos theta) = 1 - sin theta * cos theta

Answer 1

Answer on Question #82447 – Math – Trigonometry

Question

I need to establish the identity

$$(sin^3\theta + cos^3\theta) / (sin\theta + cos\theta) = 1 - sin\theta \cdot cos\theta$$

Solution

As $(x^3 + y^3) = (x + y) \cdot (x^2 - x \cdot y + y^2)$, hence

$$(sin^3\theta + cos^3\theta) = (sin\theta + cos\theta) \cdot (sin^2\theta - sin\theta \cdot cos\theta + cos^2\theta) = (sin\theta + cos\theta)(1 - sin\theta \cdot cos\theta)$$

Dividing both sides by $(sin\theta + cos\theta)$ we obtain the desired identity.

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