(a^{0.2})/((cos^2(x))-cos^2(x)-1)=0

{ "query": { "display": "$$\\frac{a^{0.2}}{\\left(\\cos^{2}\\left(x\\right)\\right)-\\cos^{2}\\left(x\\right)-1}=0$$", "symbolab_question": "EQUATION#\\frac{a^{0.2}}{(\\cos^{2}(x))-\\cos^{2}(x)-1}=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{a^{0.2}}{\\left(\\cos^{2}\\left(x\\right)\\right)-\\cos^{2}\\left(x\\right)-1}=0{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\frac{a^{0.2}}{\\left(\\cos^{2}\\left(x\\right)\\right)-\\cos^{2}\\left(x\\right)-1}=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\frac{a^{0.2}}{\\cos^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)-1}=0", "result": "\\mathrm{True\\:for\\:all}\\:\\cos\\left(x\\right);\\quad\\:-a^{0.2}=0", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "\\frac{a^{0.2}}{u^{2}-u^{2}-1}=0" }, { "type": "interim", "title": "$$\\frac{a^{0.2}}{u^{2}-u^{2}-1}=0{\\quad:\\quad}$$True for all $$u;\\quad\\:-a^{0.2}=0$$", "input": "\\frac{a^{0.2}}{u^{2}-u^{2}-1}=0", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{a^{0.2}}{u^{2}-u^{2}-1}:{\\quad}-a^{0.2}$$", "input": "\\frac{a^{0.2}}{u^{2}-u^{2}-1}", "result": "-a^{0.2}=0", "steps": [ { "type": "step", "primary": "Add similar elements: $$u^{2}-u^{2}=0$$", "result": "=\\frac{a^{0.2}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{a^{0.2}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-a^{0.2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcAVspF7pmlN67dRKIr8Iq3pEznrQqWJVtLayJzdJgQAlilG71elit3w1IBbYN0PbtmDbInmz0AWL8h3UaoVsCDmLksDnGUf3Aryv9buk6CLGmNnLPWGf9PH3lpmjoJIY7jPBBFzTT2ii74gXbU7fqyxHPoZjHBTwnnGDfjz2izgSzpqCMNAkhhkCvk7ftoB" } }, { "type": "step", "primary": "Both sides are equal" }, { "type": "step", "result": "\\mathrm{True\\:for\\:all}\\:u;\\quad\\:-a^{0.2}=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\mathrm{True\\:for\\:all}\\:\\cos\\left(x\\right);\\quad\\:-a^{0.2}=0" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=$$True for all $$u{\\quad:\\quad}x=\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn,\\:x=-\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn$$", "input": "\\cos\\left(x\\right)=\\mathrm{True\\:for\\:all}\\:u", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=\\mathrm{True\\:for\\:all}\\:u", "result": "x=\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn,\\:x=-\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=$$True for all $$u$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(a\\right)+2πn$$" ], "result": "x=\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn,\\:x=-\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn,\\:x=-\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn" }, { "type": "step", "primary": "Since the equation is undefined for:$${\\quad}\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn,\\:-\\arccos\\left(\\mathrm{True\\:for\\:all}\\:u\\right)+2πn$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{a^{0.2}}{(\\cos^{2}(x))-\\cos^{2}(x)-1}=0" }, "showViewLarger": true } }, "meta": { "showVerify": true } }