# Multiple and sub- multiple Angles

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# If $\frac{{\mathrm{sin}}^{3}\mathrm{\theta }}{\mathrm{sin}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)}=\frac{{\mathrm{cos}}^{3}\mathrm{\theta }}{\mathrm{cos}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)}$ and $\mathrm{tan}2\mathrm{\theta }=\mathrm{\lambda tan}\left(3\mathrm{\theta }+\mathrm{\alpha }\right)$, then the value of $\mathrm{\lambda }$ is _________.

Moderate
Solution

## We have$\frac{{\mathrm{sin}}^{3}\mathrm{\theta }}{\mathrm{sin}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)}=\frac{{\mathrm{cos}}^{3}\mathrm{\theta }}{\mathrm{cos}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)}=\mathrm{k}\left(\text{let}\right)$              …. (1)Also,$\frac{{\mathrm{sin}}^{3}\mathrm{\theta cos}\mathrm{\theta }}{\mathrm{sin}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)\mathrm{cos}\mathrm{\theta }}=\frac{\mathrm{sin}{\mathrm{\theta cos}}^{3}\mathrm{\theta }}{\mathrm{sin}\mathrm{\theta cos}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)}=\mathrm{k}$                $=\mathrm{k}\left(\mathrm{sin}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)\mathrm{cos}\mathrm{\theta }+\mathrm{sin}\mathrm{\theta cos}\left(2\mathrm{\theta }+\mathrm{\alpha }\right)\right)$           …… (2)From (1) and (2), we get$\mathrm{tan}2\mathrm{\theta }=2\mathrm{tan}\left(3\mathrm{\theta }+\mathrm{\alpha }\right)$

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Let $\mathrm{P}\left(\mathrm{k}\right)=\left(1+\mathrm{cos}\frac{\mathrm{\pi }}{4\mathrm{k}}\right)\left(1+\mathrm{cos}\frac{\left(2\mathrm{k}-1\right)\mathrm{\pi }}{4\mathrm{k}}\right)$ $\left(1+\mathrm{cos}\frac{\left(2\mathrm{k}+1\right)\mathrm{\pi }}{4\mathrm{k}}\right)\left(1+\mathrm{cos}\frac{\left(4\mathrm{k}-1\right)\mathrm{\pi }}{4\mathrm{k}}\right)$. Then