That night, instead of working, he searched online: Answers for No Joking Around Trigonometric Identities . He found a blurry image from two years ago—same worksheet, different school. He copied every line.
“You didn’t memorize steps. You reasoned .” She handed back his paper. “Next time, trust your own brain instead of someone else’s answer key.” Answers For No Joking Around Trigonometric Identities
Mrs. Castillo nodded. “You just derived it yourself.” That night, instead of working, he searched online:
He stood at the board, chalk in hand, sweating. He wrote (\frac{\sin x}{1+\cos x} \cdot \frac{1-\cos x}{1-\cos x}). Then (\frac{\sin x(1-\cos x)}{1-\cos^2 x}). Then (\frac{\sin x(1-\cos x)}{\sin^2 x}). Then (\frac{1-\cos x}{\sin x}). Then (\frac{1}{\sin x} - \frac{\cos x}{\sin x} = \csc x - \cot x). instead of working