{"generator":"equations.identity","version":1,"value":{"problems":[{"prompt":"cos(a - b) = cos(a)cos(b) + sin(a)sin(b)","prompt_latex":"\\cos\\left(a - b\\right) = \\cos\\left(a\\right)\\cos\\left(b\\right) + \\sin\\left(a\\right)\\sin\\left(b\\right)","answer":"True","answer_latex":"\\text{True}","holds":true,"identity":"cos-difference"},{"prompt":"1 + tan(x)^2 = 1\/cos(x)^2","prompt_latex":"1 + \\tan^{2}\\left(x\\right) = \\frac{1}{\\cos^{2}\\left(x\\right)}","answer":"True","answer_latex":"\\text{True}","holds":true,"identity":"pythagorean-tan"},{"prompt":"exp(a + b) = exp(a)ln(b)","prompt_latex":"e^{a + b} = e^{a}\\ln\\left(b\\right)","answer":"False \u2014 a exp became a ln; it should read exp(a + b) = exp(a)exp(b)","answer_latex":"\\text{False}","holds":false,"identity":"exp-sum","corruption":"a exp became a ln"},{"prompt":"cos(a + b) = cos(a)cos(b) - sin(a)sin(b)","prompt_latex":"\\cos\\left(a + b\\right) = \\cos\\left(a\\right)\\cos\\left(b\\right) - \\sin\\left(a\\right)\\sin\\left(b\\right)","answer":"True","answer_latex":"\\text{True}","holds":true,"identity":"cos-sum"}],"count":4},"display":"1. cos(a - b) = cos(a)cos(b) + sin(a)sin(b)   \u2192   True\n2. 1 + tan(x)^2 = 1\/cos(x)^2   \u2192   True\n3. exp(a + b) = exp(a)ln(b)   \u2192   False \u2014 a exp became a ln; it should read exp(a + b) = exp(a)exp(b)\n4. cos(a + b) = cos(a)cos(b) - sin(a)sin(b)   \u2192   True","meta":{"latex":[["\\cos\\left(a - b\\right) = \\cos\\left(a\\right)\\cos\\left(b\\right) + \\sin\\left(a\\right)\\sin\\left(b\\right)","\\text{True}"],["1 + \\tan^{2}\\left(x\\right) = \\frac{1}{\\cos^{2}\\left(x\\right)}","\\text{True}"],["e^{a + b} = e^{a}\\ln\\left(b\\right)","\\text{False}"],["\\cos\\left(a + b\\right) = \\cos\\left(a\\right)\\cos\\left(b\\right) - \\sin\\left(a\\right)\\sin\\left(b\\right)","\\text{True}"]],"problems":4,"topic":"mixed","true_statements":3,"stream_bytes_used":33,"rejections":0,"entropy_in_bytes":16,"duration_us":880},"receipt":{"source":"atmosphere","source_label":"Live Weather","class":"C","class_label":"Observational","caveat":"Carries only tens of bits of genuine surprise. Always mixed with the OS CSPRNG.","narrative":"It was 2.1\u00b0C in Yakutsk, Russia, with a 16 km\/h wind, when this was made.","proof_url":"https:\/\/open-meteo.com\/en\/docs#latitude=62.035&longitude=129.675","reference":"Yakutsk, Russia","observed_at":"2026-09-13T19:28:36+00:00","degraded":false,"mixed_with_csprng":true,"latency_ms":451,"cached":false},"seed":{"token":"S096MWF39DAX5M6MJ9Y0R15Y8C","permalink":"https:\/\/randomly.cluxnei.dev\/r\/S096MWF39DAX5M6MJ9Y0R15Y8C?g=equations.identity&v=1&p=eyJjb3VudCI6NCwiZXhwbGFpbiI6dHJ1ZSwidG9waWMiOiJtaXhlZCJ9&s=atmosphere&r=Yakutsk%2C%20Russia"},"uses_entropy":true,"reproducible":true}