evaluate cossec^(-1)(5/3). give your answer whenever necessary.

Explanation:

Step1: Recall the inverse - secant relationship

Let $\theta=\sec^{-1}\left(\frac{5}{3}\right)$. Then, by the definition of the inverse secant function, $\sec\theta=\frac{5}{3}$, where $\theta\in\left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right]$.

Step2: Use the reciprocal identity

Since $\sec\theta=\frac{1}{\cos\theta}=\frac{5}{3}$, we can solve for $\cos\theta$. Cross - multiplying gives $\cos\theta=\frac{3}{5}$. So, $\cos\left[\sec^{-1}\left(\frac{5}{3}\right)\right]=\frac{3}{5}$.

Answer:

$\frac{3}{5}$