Question

The period of the function: $f\left(x\right)=e^{\ln \left(\sin x\right)}+{\tan }^{3}x-\text{cosec}(3x-5)$ is $b\pi$. Find $b$

Answer 1

$f\left(x\right)=e^{\ln \left(\sin x\right)}+{\tan }^{3}x-\text{cosec}(3x-5)=\sin x+{\tan }^{3}x-\text{cosec}(3x-5)$
Period of $\sin x,{\tan }^{3}x$ and $\text{cosec}(3x-5)$ are $2\pi ,\pi$ and $\frac{2\pi }{3}$
$\Rightarrow$ Period of $f\left(x\right)$ is $=$ L.C.M of $2\pi ,\pi$ and $\frac{2\pi }{3}=2\pi$
Hence $b=2$