Find the period of the function

Q(viii) f (x) = e^{In (sin x)} + tan³ x – cosec (3x – 5)

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\ln \left\{\mathrm{sinx}\right\}}+{\tan }^3\mathrm{x}-\mathrm{cosec}\left(3\mathrm{x}-5\right) \\ \mathrm{Period} \mathrm{of} \sin \mathrm{x} \mathrm{is} 2\mathrm{\pi }, \mathrm{hence} {\mathrm{e}}^{\ln \left\{\mathrm{sinx}\right\}} \mathrm{will} \mathrm{also} \mathrm{have} \mathrm{period} 2\mathrm{\pi } \\ \mathrm{Period} \mathrm{of} {\tan }^{\mathrm{n}}\left(\mathrm{x}\right) \left(\mathrm{n}\in \mathrm{N}\right) \mathrm{is} \mathrm{\pi }, \mathrm{hence} {\tan }^3\mathrm{x} \mathrm{will} \mathrm{have} \mathrm{period} \mathrm{\pi } \\ \mathrm{Period} \mathrm{of} \mathrm{cosec}\left(\mathrm{ax}+\mathrm{b}\right) \mathrm{is} \frac{2\mathrm{\pi }}{\mathrm{a}}, \mathrm{hence} \mathrm{cosec}\left(3\mathrm{x}-5\right) \mathrm{will} \mathrm{have} \mathrm{period} \frac{2\mathrm{\pi }}{3} \\ \mathrm{Hence} \mathrm{period} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \\ \mathrm{L}.\mathrm{C}.\mathrm{M}.\left\{2\mathrm{\pi },\mathrm{\pi },\frac{2\mathrm{\pi }}{3}\right\}=\frac{\mathrm{L}.\mathrm{C}.\mathrm{M}.\left\{2\mathrm{\pi },\mathrm{\pi },2\mathrm{\pi }\right\}}{\mathrm{H}.\mathrm{C}.\mathrm{F}.\left\{1,1,3\right\}}=\frac{2\mathrm{\pi }}{1}=2\mathrm{\pi } \end{gathered}$$

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