{"generator":"equations.identity","version":1,"value":{"problems":[{"prompt":"cos(2x) = cos(x)^2 - sin(x)^2","prompt_latex":"\\cos\\left(2x\\right) = \\cos^{2}\\left(x\\right) - \\sin^{2}\\left(x\\right)","answer":"True","answer_latex":"\\text{True}","holds":true,"identity":"cos-double-difference"},{"prompt":"ln(a\/b) = exp(a) - ln(b)","prompt_latex":"\\ln\\left(\\frac{a}{b}\\right) = e^{a} - \\ln\\left(b\\right)","answer":"False \u2014 a ln became a exp; it should read ln(a\/b) = ln(a) - ln(b)","answer_latex":"\\text{False}","holds":false,"identity":"log-quotient","corruption":"a ln became a exp"},{"prompt":"2sin(a)cos(b) = sin(a + b) + sin(a - b)","prompt_latex":"2\\sin\\left(a\\right)\\cos\\left(b\\right) = \\sin\\left(a + b\\right) + \\sin\\left(a - b\\right)","answer":"True","answer_latex":"\\text{True}","holds":true,"identity":"product-to-sum"},{"prompt":"exp(a)^2 = exp(2a)","prompt_latex":"\\left(e^{a}\\right)^{2} = e^{2a}","answer":"True","answer_latex":"\\text{True}","holds":true,"identity":"exp-power"}],"count":4},"display":"1. cos(2x) = cos(x)^2 - sin(x)^2   \u2192   True\n2. ln(a\/b) = exp(a) - ln(b)   \u2192   False \u2014 a ln became a exp; it should read ln(a\/b) = ln(a) - ln(b)\n3. 2sin(a)cos(b) = sin(a + b) + sin(a - b)   \u2192   True\n4. exp(a)^2 = exp(2a)   \u2192   True","meta":{"latex":[["\\cos\\left(2x\\right) = \\cos^{2}\\left(x\\right) - \\sin^{2}\\left(x\\right)","\\text{True}"],["\\ln\\left(\\frac{a}{b}\\right) = e^{a} - \\ln\\left(b\\right)","\\text{False}"],["2\\sin\\left(a\\right)\\cos\\left(b\\right) = \\sin\\left(a + b\\right) + \\sin\\left(a - b\\right)","\\text{True}"],["\\left(e^{a}\\right)^{2} = e^{2a}","\\text{True}"]],"problems":4,"topic":"mixed","true_statements":3,"stream_bytes_used":34,"rejections":1,"entropy_in_bytes":16,"duration_us":602},"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 14.4\u00b0C in Ushuaia, Argentina, with a 6 km\/h wind, when this was made.","proof_url":"https:\/\/open-meteo.com\/en\/docs#latitude=-54.801&longitude=-68.303","reference":"Ushuaia, Argentina","observed_at":"2026-10-28T23:08:36+00:00","degraded":false,"mixed_with_csprng":true,"latency_ms":422,"cached":false},"seed":{"token":"BS6Y5W7MWPCXDVX3M1ACP3G8SR","permalink":"https:\/\/randomly.cluxnei.dev\/r\/BS6Y5W7MWPCXDVX3M1ACP3G8SR?g=equations.identity&v=1&p=eyJjb3VudCI6NCwiZXhwbGFpbiI6dHJ1ZSwidG9waWMiOiJtaXhlZCJ9&s=atmosphere&r=Ushuaia%2C%20Argentina"},"uses_entropy":true,"reproducible":true}