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Prove that this function proves the relation written next to it. $y=\sin(x)+a\cdot \cos(x)$ ...the relation is $ y\cdot \sin(x) +y'\cdot \cos(x)=1$

I tried using logarithmic differentiation but nothing...

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You don't need logarithmic differentation for this. You have that $y =\sin x + a \cos x$. First of all, you should find $y'$, which I assume you know how to do. Now multiply $y$ by $\sin x$, $y'$ by $\cos x$ and add them up.

I'll help you with the first one: $y \sin x = \sin^2 x + a \cos x \sin x$. After finding $y'$, multiply that by $\cos x$ and add it to what we just got. The result should be $1$.

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