What is the relationship between secant and cosecant in terms of acute angles?

[karush]

If
$$\tan\left({\alpha}\right)\tan\left({\beta}\right)=1$$
$\alpha$ and $\beta$ are acute angles

Then
$$\sec\left({\alpha}\right)=\csc\left({\beta}\right)$$

Again there's options, I tried the product to sum formulas but it went off in a bad direction

 

[MarkFL]

I would rewrite the given equation as:

\(\displaystyle \tan(\alpha)=\cot(\beta)\)

So, we can now easily see that $\alpha$ and $\beta$ are complementary. And so the result we are to prove follows, since secant and cosecant are complementary functions by definition. :)

 

[Greg]

\(\displaystyle \tan\alpha\tan\beta=1\)

\(\displaystyle \tan^2\alpha\tan^2\beta=1\)

\(\displaystyle (\sec^2\alpha-1)\tan^2\beta=1\)

\(\displaystyle \sec^2\alpha-1=\cot^2\beta\)

\(\displaystyle \sec^2\alpha=\cot^2\beta+1\)

\(\displaystyle \sec^2\alpha=\csc^2\beta\)

\(\displaystyle \sec\alpha=\csc\beta\)