If f(x)=(cos x)(cos 2 x) ldots(cos n x),then | vedclass

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(B) Given $f(x) = \prod_{r=1}^n \cos(rx)$.Taking the natural logarithm on both sides:$\ln|f(x)| = \sum_{r=1}^n \ln|\cos(rx)|$.Differentiating both sid

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(B) Given $f(x) = \prod_{r=1}^n \cos(rx)$.Taking the natural logarithm on both sides:$\ln|f(x)| = \sum_{r=1}^n \ln|\cos(rx)|$.Differentiating both sides with respect to $x$:$\frac{1}{f(x)} f^{\prime}(x) = \sum_{r=1}^n \frac{1}{\cos(rx)} \cdot (-\sin(rx) \cdot r)$.$\frac{f^{\prime}(x)}{f(x)} = -\sum_{r=1}^n r 	an(rx)$.Multiplying both sides by $f(x)$:$f^{\prime}(x) = -f(x) \sum_{r=1}^n r 	an(rx)$.Rearranging the terms:$f^{\prime}(x) + \sum_{r=1}^n (r 	an(rx)) f(x) = 0$.

If $f(x)=(\cos x)(\cos 2 x) \ldots(\cos n x)$,then $f^{\prime}(x)+\sum_{r=1}^n(r \tan r x) f(x)$ is equal to

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