// Numbas version: finer_feedback_settings
{"name": "Trigonometry: Simplifying Trigonometric Expressions 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry: Simplifying Trigonometric Expressions 1", "tags": [], "metadata": {"description": "<p>Simplifying the trigonometric expression $\\frac{\\sin^2(x)}{1\\pm \\cos(x)}$ using the trigonometric identity $\\sin^2(x)+\\cos^2(x)=1$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Using the identity $\\sin^2(x)+\\cos^2(x) = 1$, simplify the following expression:</p>\n<p>\\[ \\frac{\\sin^2(x)}{\\simplify{1+{a}cos(x)}} .\\]</p>", "advice": "<p>To simplify the expression $\\frac{\\sin^2(x)}{\\simplify{1+{a}cos(x)}}$, we want to start by rewriting the numerator in terms of $\\cos(x)$. This can be done by rearranging the trigonometric identity:</p>\n<p>\\[ \\sin^2(x)+\\cos^2(x)=1 \\implies \\sin^2(x) = 1-\\cos^2(x).\\]</p>\n<p>Hence,</p>\n<p>\\[ \\frac{\\sin^2(x)}{\\simplify{1+{a}cos(x)}} = \\frac{1-\\cos^2(x)}{\\simplify{1+{a}cos(x)}}.\\]</p>\n<p>Then using the difference of two squares, we can simplify our expression further:</p>\n<p>\\[ \\begin{split} \\frac{1-\\cos^2(x)}{\\simplify{1+{a}cos(x)}} &amp;\\,= \\frac{(1-\\cos(x))(1+\\cos(x))}{\\simplify{1+{a}cos(x)}} \\\\\\\\ &amp;\\,= \\simplify{1-{a}cos(x)}. \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\frac{\\sin^2(x)}{\\simplify{1+{a}cos(x)}} = \\simplify{1-{a}cos(x)}.\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1-{a}cos(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}