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

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The relationship between secant and cosecant for acute angles is established through the equation tan(α)tan(β) = 1, indicating that α and β are complementary angles. This leads to the conclusion that sec(α) = csc(β), as secant and cosecant are defined as complementary functions. The discussion also explores the mathematical transformations that confirm this relationship, including rewriting tan(α) as cot(β) and deriving sec²(α) = csc²(β). The proof demonstrates the interconnectedness of these trigonometric functions in the context of acute angles. Understanding this relationship is essential for solving various trigonometric problems.
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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
 
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I would rewrite the given equation as:

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

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. :)
 
$$\tan\alpha\tan\beta=1$$

$$\tan^2\alpha\tan^2\beta=1$$

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

$$\sec^2\alpha-1=\cot^2\beta$$

$$\sec^2\alpha=\cot^2\beta+1$$

$$\sec^2\alpha=\csc^2\beta$$

$$\sec\alpha=\csc\beta$$
 
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