csc(x)(cos^2(x)+sqrt(2))=0

{ "query": { "display": "$$\\csc\\left(x\\right)\\left(\\cos^{2}\\left(x\\right)+\\sqrt{2}\\right)=0$$", "symbolab_question": "EQUATION#\\csc(x)(\\cos^{2}(x)+\\sqrt{2})=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": "$$\\csc\\left(x\\right)\\left(\\cos^{2}\\left(x\\right)+\\sqrt{2}\\right)=0{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\csc\\left(x\\right)\\left(\\cos^{2}\\left(x\\right)+\\sqrt{2}\\right)=0", "steps": [ { "type": "step", "primary": "Solving each part separately", "result": "\\csc\\left(x\\right)=0\\lor\\:\\cos^{2}\\left(x\\right)+\\sqrt{2}=0" }, { "type": "interim", "title": "$$\\csc\\left(x\\right)=0{\\quad:\\quad}$$No Solution", "input": "\\csc\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)\\le-1\\lor\\csc\\left(x\\right)\\ge1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cos^{2}\\left(x\\right)+\\sqrt{2}=0{\\quad:\\quad}$$No Solution", "input": "\\cos^{2}\\left(x\\right)+\\sqrt{2}=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\cos^{2}\\left(x\\right)+\\sqrt{2}=0", "result": "\\cos\\left(x\\right)=\\sqrt[4]{2}i,\\:\\cos\\left(x\\right)=-\\sqrt[4]{2}i", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "u^{2}+\\sqrt{2}=0" }, { "type": "interim", "title": "$$u^{2}+\\sqrt{2}=0{\\quad:\\quad}u=\\sqrt[4]{2}i,\\:u=-\\sqrt[4]{2}i$$", "input": "u^{2}+\\sqrt{2}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{2}\\:$$to the right side", "input": "u^{2}+\\sqrt{2}=0", "result": "u^{2}=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{2}$$ from both sides", "result": "u^{2}+\\sqrt{2}-\\sqrt{2}=0-\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=-\\sqrt{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{-\\sqrt{2}},\\:u=-\\sqrt{-\\sqrt{2}}" }, { "type": "interim", "title": "Simplify $$\\sqrt{-\\sqrt{2}}:{\\quad}\\sqrt[4]{2}i$$", "input": "\\sqrt{-\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{-1}\\sqrt{a}$$", "secondary": [ "$$\\sqrt{-\\sqrt{2}}=\\sqrt{-1}\\sqrt{\\sqrt{2}}$$" ], "result": "=\\sqrt{-1}\\sqrt{\\sqrt{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=i\\sqrt{\\sqrt{2}}", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sqrt{\\sqrt{2}}:{\\quad}\\sqrt[4]{2}$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\frac{1}{2}}\\right)^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=2^{\\frac{1}{2}\\cdot\\:\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:\\frac{1}{2}=\\frac{1}{4}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{1}{2}", "result": "=2^{\\frac{1}{4}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8auvd86dSXYkWbbgR3KJLXB8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcauvogRRA/ie/KMYc4sBzhqZvNWyGcX6HZt1LGXH2QGa+LZe+3IobTh1qApFsHX0PdPD78SM8rC5Gb0Tet2Vbph2NOtqSMcWJbA1OsBAA1UpdB" } }, { "type": "step", "primary": "$$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "result": "=\\sqrt[4]{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=i\\sqrt[4]{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$-\\sqrt{-\\sqrt{2}}:{\\quad}-\\sqrt[4]{2}i$$", "input": "-\\sqrt{-\\sqrt{2}}", "steps": [ { "type": "interim", "title": "Simplify $$\\sqrt{-\\sqrt{2}}:{\\quad}\\sqrt[4]{2}i$$", "input": "\\sqrt{-\\sqrt{2}}", "result": "=-\\sqrt[4]{2}i", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{-1}\\sqrt{a}$$", "secondary": [ "$$\\sqrt{-\\sqrt{2}}=\\sqrt{-1}\\sqrt{\\sqrt{2}}$$" ], "result": "=\\sqrt{-1}\\sqrt{\\sqrt{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=i\\sqrt{\\sqrt{2}}", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sqrt{\\sqrt{2}}:{\\quad}\\sqrt[4]{2}$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\frac{1}{2}}\\right)^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=2^{\\frac{1}{2}\\cdot\\:\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:\\frac{1}{2}=\\frac{1}{4}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{1}{2}", "result": "=2^{\\frac{1}{4}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8auvd86dSXYkWbbgR3KJLXB8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcauvogRRA/ie/KMYc4sBzhqZvNWyGcX6HZt1LGXH2QGa+LZe+3IobTh1qApFsHX0PdPD78SM8rC5Gb0Tet2Vbph2NOtqSMcWJbA1OsBAA1UpdB" } }, { "type": "step", "primary": "$$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "result": "=\\sqrt[4]{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=i\\sqrt[4]{2}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "u=\\sqrt[4]{2}i,\\:u=-\\sqrt[4]{2}i" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=\\sqrt[4]{2}i,\\:\\cos\\left(x\\right)=-\\sqrt[4]{2}i" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\sqrt[4]{2}i{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=\\sqrt[4]{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-\\sqrt[4]{2}i{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=-\\sqrt[4]{2}i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "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": "\\csc(x)(\\cos^{2}(x)+\\sqrt{2})" }, "showViewLarger": true } }, "meta": { "showVerify": true } }