(Cos to the power of 3 A - sin to the power of 3 A) / cos A - sin A = (2 + sin 2A) /2

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Solution

$LHS = \frac{\cos^{3} A - \sin^{3} A}{(\cos A - \sin A)} = \frac{{(\cos A)}^{3} - {(\sin A)}^{3}}{(\cos A - \sin A)} = \frac{(\cos A - \sin A) (\cos^{2} A + \sin^{2} A + \cos A . \sin A)}{(\cos A - \sin A)} = (\cos^{2} A + \sin^{2} A + \cos A . \sin A) = (\cos^{2} A + \sin^{2} A) + \cos A . \sin A = 1 + \cos A . \sin A RHS = \frac{2 + \sin 2 A}{2} = \frac{2 + 2 \sin A . \cos A}{2} = \frac{2 (1 + \sin A . \cos A)}{2} = 1 + \cos A . \sin A Hence, LHS = RHS$

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