+ (t^2 - 2) + 2t = t^2 + 2t + 1 = (t + 1)^2

February 22, 2026

+ (t^2 - 2) + 2t = t^2 + 2t + 1 = (t + 1)^2

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+ (1 + \tan^2 x) + (1 + \cot^2 x) + 2(\tan x + \cot x) = 3 + \tan^2 x + \cot^2 x + 2(\tan x + \cot x)
Let $ t = \tan x + \cot x $. Then:
\tan^2 x + \cot^2 x = t^2 - 2
Now, $ t = \tan x + \cot x \geq 2 $ by AM-GM, with equality when $ \tan x = 1 \Rightarrow x = \frac{\pi}{4} $.
So minimum occurs at $ t = 2 $, giving:
$$Question: A right triangular solar panel on a spacecraft has a hypotenuse of length $ z $ and an inscribed circle with radius $ c $. What is the ratio of the area of the inscribed circle to the area of the triangle?

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