// Numbas version: finer_feedback_settings
{"name": "Trig identity 9", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trig identity 9", "tags": [], "metadata": {"description": "<p>Students need to prove a given trig identity, Written as a scan solution and upload question, no randomisation.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Show that the identity \\[\\cos 2\\theta=\\frac{1-\\tan^2\\theta}{1+\\tan^2\\theta}\\] is true. Write your proof in your handwritten working and upload it.</p>", "advice": "<p>\\begin{align}RHS&amp;=\\frac{1-\\tan^2\\theta}{1+\\tan^2\\theta}\\\\&amp;=\\frac{\\frac{\\cos^2\\theta-\\sin^2\\theta}{\\cos^2\\theta}}{\\frac{\\cos^2\\theta+\\sin^2\\theta}{\\cos^2\\theta}}\\quad\\text{using the identity for }\\tan\\theta\\\\&amp;=\\frac{\\cos^2\\theta-\\sin^2\\theta}{\\cos^2\\theta}\\times \\cos^2\\theta\\quad\\text{fraction propetries and the Pythagorean identity }\\\\&amp;=\\cos^2\\theta-\\sin^2\\theta\\quad\\text{simplification}\\\\&amp;=\\cos 2\\theta\\quad\\text{from standard formula for }\\cos 2\\theta\\\\&amp;=LHS\\end{align}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}]}]}], "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}]}