Can someone kindly explain? Thanks

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The discussion revolves around the confusion regarding the transformation of arcsec to arctan in a trigonometric context. The user describes constructing a right triangle to visualize the relationship between angles α and β, where α is defined using secant and β using tangent. The key point is that the relationship between these angles can be established through the definitions of secant and tangent in relation to the triangle's sides. The user seeks clarification on how these angles relate to each other mathematically. Understanding this relationship is crucial for grasping the underlying concepts in trigonometric transformations.
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Homework Statement


I have attached my question. I don't understand how A can turn into B? (the highlighted parts)


Homework Equations





The Attempt at a Solution


I finished the whole question, but I don't understand why arcsec turns into arctan...
 

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Draw a right triangle with acute angle α at the left and another acute angle β at the upper right. Label the hypotenuse as x and the base as 4. Then the altitude is sqrt(x2 - 16).

Then x/4 = sec(α), so α = sec-1(x/4).

And 4/sqrt(x2 - 16) = tan(β), so β = tan-1( 4/sqrt(x2 - 16)).

What can you say about the angles α and β?
 

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