If frac{cos 3A + sin 3A}{cos A - sin A} = 1 - | vedclass

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(A) Given expression: $\frac{\cos 3A + \sin 3A}{\cos A - \sin A} = 1 - K \sin 2A$Using the identities $\cos 3A = 4 \cos^3 A - 3 \cos A$ and $\sin 3A =

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$-2$

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(A) Given expression: $\frac{\cos 3A + \sin 3A}{\cos A - \sin A} = 1 - K \sin 2A$Using the identities $\cos 3A = 4 \cos^3 A - 3 \cos A$ and $\sin 3A = 3 \sin A - 4 \sin^3 A$:Numerator $= (4 \cos^3 A - 3 \cos A) + (3 \sin A - 4 \sin^3 A)$$= 4(\cos^3 A - \sin^3 A) - 3(\cos A - \sin A)$$= 4(\cos A - \sin A)(\cos^2 A + \sin^2 A + \sin A \cos A) - 3(\cos A - \sin A)$$= 4(\cos A - \sin A)(1 + \sin A \cos A) - 3(\cos A - \sin A)$Dividing by $(\cos A - \sin A)$:$= 4(1 + \sin A \cos A) - 3$$= 4 + 4 \sin A \cos A - 3$$= 1 + 2(2 \sin A \cos A)$$= 1 + 2 \sin 2A$Comparing this with $1 - K \sin 2A$,we get $-K = 2$,so $K = -2$.

If $\frac{\cos 3A + \sin 3A}{\cos A - \sin A} = 1 - K \sin 2A,$ the value of $K$ is

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