$$ an(sec^-1 4)$$

without making use of a calculator (you need to discover the precise value)?

How to proceed?

Imagine a right-angled triangle through one leg $k$ and hypotenuse $4k$ and angle $ heta$ in between them. Then $cos heta = frack4k= frac14$ and $sec heta = 4$, making $sec^-14 = heta$.

The opposite leg is $sqrt(4k)^2-k^2=sqrt15k$ and so $ an(sec^-14) = an heta = fracsqrt15kk=sqrt15$. Now you might need a calculator.

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Let $sec^-14= hetaimpliessec heta=4$

Now, $ an^2 heta=sec^2 heta-1$

Finally utilizing the definition of the primary value of $sec^-1,00$

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