# Trigonometric transformations

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# Given that, $\mathrm{f}\left(\mathrm{n\theta }\right)=\frac{2\mathrm{sin}2\mathrm{\theta }}{\mathrm{cos}2\mathrm{\theta }-\mathrm{cos}4\mathrm{n\theta }}$, and $\mathrm{f}\left(\mathrm{\theta }\right)+\mathrm{f}\left(2\mathrm{\theta }\right)+\mathrm{f}\left(3\mathrm{\theta }\right)+\dots +\mathrm{f}\left(\mathrm{n\theta }\right)=\frac{\mathrm{sin}\mathrm{\lambda \theta }}{\mathrm{sin}\mathrm{\theta sin}\mathrm{\mu \theta }}$, then the value of $\mathrm{\mu }-\mathrm{\lambda }$ is __________.

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Solution

## $\begin{array}{c}\mathrm{f}\left(\mathrm{n\theta }\right)=\frac{2\mathrm{sin}2\mathrm{\theta }}{\mathrm{cos}2\mathrm{\theta }-\mathrm{cos}4\mathrm{n\theta }}\\ =\frac{2\mathrm{sin}2\mathrm{\theta }}{2\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\left(2\mathrm{n}-1\right)\mathrm{\theta }}\\ =\frac{\mathrm{sin}\left(\left(2\mathrm{n}+1\right)\mathrm{\theta }-\left(2\mathrm{n}-1\right)\mathrm{\theta }\right)}{\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\left(2\mathrm{n}-1\right)\mathrm{\theta }}\\ =\frac{\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta cos}\left(2\mathrm{n}-1\right)\mathrm{\theta }-\mathrm{cos}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\left(2\mathrm{n}-1\right)\mathrm{\theta }}{\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\left(2\mathrm{n}-1\right)\mathrm{\theta }}\\ =\mathrm{cot}\left(2\mathrm{n}-1\right)\mathrm{\theta }-\mathrm{cot}\left(2\mathrm{n}+1\right)\mathrm{\theta }\end{array}$        $\begin{array}{l}=\mathrm{cot}\mathrm{\theta }-\mathrm{cot}\left(2\mathrm{n}+1\right)\mathrm{\theta }\\ =\frac{\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta cos}\mathrm{\theta }-\mathrm{cos}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\mathrm{\theta }}{\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\mathrm{\theta }}\\ =\frac{\mathrm{sin}2\mathrm{n\theta }}{\mathrm{sin}\left(2\mathrm{n}+1\right)\mathrm{\theta sin}\mathrm{\theta }}\end{array}$

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Let  be non-zero real numbers such that $2\left(\mathrm{cos}\mathrm{\beta }-\mathrm{cos}\mathrm{\alpha }\right)+\mathrm{cos}\mathrm{\alpha cos}\mathrm{\beta }=1$. Then which of the following is/are true?