Question

prove this when all angles are acute angles​

Answer 1

Step-by-step explanation:

We use the following identity:

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

$\text{Here, a = } \sin \theta \text{ and b =} \cos \theta$

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

$\\\\\text{We know that, } \\\sin^2\theta + \cos^2\theta = 1$

$\text{Therefore, }\\\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta + \cos ^2 \theta - \sin\theta\cos\theta)\\\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(1 - \sin\theta\cos\theta )$

$\text{Given expression, }\\\dfrac{\sin^3\theta + \cos^3\theta}{\sin\theta + \cos\theta} = \dfrac{(\sin\theta + \cos\theta)(1- \sin\theta\cos\theta)}{\sin\theta + \cos\theta} = 1- \sin\theta\cos\theta$

Hence proved.

Hope this helps!

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