`sec^2(pi/2 - x) - 1 = cot^2(x)` Verfiy the identity.

Textbook Question

Chapter 5, 5.2 - Problem 46 - Precalculus (3rd Edition, Ron Larson).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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First, `sec(x) = 1/cos(x)` and `cot(x) = cos(x)/sin(x).`

Second, `cos(pi/2-x) = sin(x)` and `sin(pi/2-x) = cos(x).`

Therefore the left part is

`1/(cos^2(pi/2-x)) - 1 = 1/(sin^2(x)) - 1 = (1-sin^2(x))/(sin^2(x)) = (cos^2(x))/(sin^2(x)) = cot^2(x),` QED.

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balajia | College Teacher | (Level 1) eNoter

Posted on

`sec^2(pi/2-x) = csc^2(x)`

`sec^2(pi/2-x)-1 = csc^2x-1`

`=csc^2x-(csc^2x-cot^2x) = cot^2x` . Since `csc^2x-cot^2x=1.`

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