Alright, beta, picture this: start with the left side, 1 + tan²A. Write tan²A as sin²A/cos²A, so it becomes 1 + (sin²A/cos²A). Combine into (cos²A + sin²A)/cos²A. See? The top is just 1 from our old friend sin²A + cos²A = 1, and below is cos²A. So it becomes 1/cos²A, which is sec²A—done.
Honestly, when I first saw this proof years ago in my attic, it felt like a mountain, just like my father’s old post-office ledgers. But the trick is to start where you are, with what you know—like he sorted letters one village at a time—and the rest follows.
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