((1+cot^2(x)))/(cos^2(x))=cot^2(x)

{ "query": { "display": "$$\\frac{\\left(1+\\cot^{2}\\left(x\\right)\\right)}{\\cos^{2}\\left(x\\right)}=\\cot^{2}\\left(x\\right)$$", "symbolab_question": "EQUATION#\\frac{(1+\\cot^{2}(x))}{\\cos^{2}(x)}=\\cot^{2}(x)" }, "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": "$$\\frac{\\left(1+\\cot^{2}\\left(x\\right)\\right)}{\\cos^{2}\\left(x\\right)}=\\cot^{2}\\left(x\\right){\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\frac{\\left(1+\\cot^{2}\\left(x\\right)\\right)}{\\cos^{2}\\left(x\\right)}=\\cot^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cot^{2}\\left(x\\right)$$ from both sides", "result": "\\frac{1+\\cot^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}-\\cot^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Simplify $$\\frac{1+\\cot^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}-\\cot^{2}\\left(x\\right):{\\quad}\\frac{1+\\cot^{2}\\left(x\\right)-\\cot^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "\\frac{1+\\cot^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}-\\cot^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\cot^{2}\\left(x\\right)=\\frac{\\cot^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "result": "=\\frac{1+\\cot^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}-\\frac{\\cot^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1+\\cot^{2}\\left(x\\right)-\\cot^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{1+\\cot^{2}\\left(x\\right)-\\cot^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}=0" }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "1+\\cot^{2}\\left(x\\right)-\\cot^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "1+\\cot^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)", "result": "\\csc^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1+\\cot^{2}\\left(x\\right)=\\csc^{2}\\left(x\\right)$$", "result": "=-\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)+\\csc^{2}\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AnIq3EpDA43ydxBwz5FMdYVxo3/C392bA8GpTYGpprnsMVOTDZTnTUuhp933dAOwFDi0mzKMer0lpqKOiQ+UcOqA3hQRF3SPC1b/ULDThaEWnK5ty7CdWiIyTeuQJVMXiQyrLvkrqkABPwTf/JlykwPl054PUYWrqmWV2/SSB9zsBrAJ/pnft4dsp/ie1DdN7Y57hLhSptqnxFfeNYvWMb10NmRcKzn4LGDVg28k09R73JgXZDmpK/eAOUtCgEWOPnIZQDCBrClhMf9W+2De8Q==" } }, { "type": "interim", "title": "Factor $$\\csc^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right):{\\quad}\\left(\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)\\left(\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)$$", "input": "\\csc^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "Rewrite $$\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)$$ as $$\\left(\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)^{2}$$", "input": "\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)\\cot^{2}\\left(x\\right)=\\left(\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)^{2}$$" ], "result": "=\\left(\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "result": "=\\csc^{2}\\left(x\\right)-\\left(\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\csc^{2}\\left(x\\right)-\\left(\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)^{2}=\\left(\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)\\left(\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)$$" ], "result": "=\\left(\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)\\left(\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\left(\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)\\left(\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)=0\\lor\\:\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)=0" }, { "type": "interim", "title": "$$\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)=0{\\quad:\\quad}$$No Solution", "input": "\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Express with sin, cos", "input": "\\csc\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)", "result": "\\frac{1+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\csc\\left(x\\right)=\\frac{1}{\\sin\\left(x\\right)}$$", "result": "=\\frac{1}{\\sin\\left(x\\right)}+\\cos\\left(x\\right)\\cot\\left(x\\right)" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\cot\\left(x\\right)=\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=\\frac{1}{\\sin\\left(x\\right)}+\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sin\\left(x\\right)}+\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}:{\\quad}\\frac{1+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\frac{1}{\\sin\\left(x\\right)}+\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "result": "=\\frac{1+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}=\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{1+1}\\left(x\\right)$$" ], "result": "=\\cos^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\cos^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw47oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJVxXBxD1gYRAlNp97nQuTZFXRu5R8U1G8Rh9s+llHwfqtic1bCnH3jLV3vr22vWk8gIJE6eFSdaQPkT4FMktmcw==" } }, { "type": "step", "result": "=\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BomPtzG+UOWoCHKw/+Shmm5AZDqQW6CzLlif7pmGqWo8h+u5fV4iBAkECPyXf7J5zMFYmi1F5Hg/ibpEToVnYwDmEzFfqMSLndLqC5qAp4dwPIzt1oWxMWVTfovlFJboZEt3ZXAiqUE0HIXrrrezJH7g6cH2OfSl85iIuybAtg4M9I0lFOxOZYaH32mNX0s3xGpk+gGxnOLtRja0BBQnbsUslKYonmvF+wmNIbKyxLWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{1}{\\sin\\left(x\\right)}+\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Express Sin Cos 0Eq" } }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "1+\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Solve by substitution", "input": "1+\\cos^{2}\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=i,\\:\\cos\\left(x\\right)=-i", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "1+u^{2}=0" }, { "type": "interim", "title": "$$1+u^{2}=0{\\quad:\\quad}u=i,\\:u=-i$$", "input": "1+u^{2}=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1+u^{2}=0", "result": "u^{2}=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1+u^{2}-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=-1" } ], "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{-1},\\:u=-\\sqrt{-1}" }, { "type": "interim", "title": "Simplify $$\\sqrt{-1}:{\\quad}i$$", "input": "\\sqrt{-1}", "steps": [ { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=i", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$-\\sqrt{-1}:{\\quad}-i$$", "input": "-\\sqrt{-1}", "steps": [ { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=-i", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "u=i,\\:u=-i" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=i,\\:\\cos\\left(x\\right)=-i" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=i{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=i", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-i{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=-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": "interim", "title": "$$\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)=0{\\quad:\\quad}$$No Solution", "input": "\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Express with sin, cos", "input": "\\csc\\left(x\\right)-\\cos\\left(x\\right)\\cot\\left(x\\right)", "result": "\\frac{1-\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\csc\\left(x\\right)=\\frac{1}{\\sin\\left(x\\right)}$$", "result": "=\\frac{1}{\\sin\\left(x\\right)}-\\cos\\left(x\\right)\\cot\\left(x\\right)" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\cot\\left(x\\right)=\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=\\frac{1}{\\sin\\left(x\\right)}-\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sin\\left(x\\right)}-\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}:{\\quad}\\frac{1-\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\frac{1}{\\sin\\left(x\\right)}-\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "result": "=\\frac{1-\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}=\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{1+1}\\left(x\\right)$$" ], "result": "=\\cos^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\cos^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw47oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJVxXBxD1gYRAlNp97nQuTZFXRu5R8U1G8Rh9s+llHwfqtic1bCnH3jLV3vr22vWk8gIJE6eFSdaQPkT4FMktmcw==" } }, { "type": "step", "result": "=\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BomPtzG+UOWoCHKw/+Shmm5AZDqQW6CzLlif7pmGqWo8h+u5fV4iBAkECPyXf7J5zMFYmi1F5Hg/ibpEToVnYwDmEzFfqMSLndLqC5qAp4dwPIzt1oWxMWVTfovlFJboZEt3ZXAiqUE0HIXrrrezJH7g6cH2OfSl85iIuybAtg4M9I0lFOxOZYaH32mNX0s3xGpk+gGxnOLtRja0BBQnbsUslKYonmvF+wmNIbKyxLWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{1}{\\sin\\left(x\\right)}-\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1-\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Express Sin Cos 0Eq" } }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "1-\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Solve by substitution", "input": "1-\\cos^{2}\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=1,\\:\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "1-u^{2}=0" }, { "type": "interim", "title": "$$1-u^{2}=0{\\quad:\\quad}u=1,\\:u=-1$$", "input": "1-u^{2}=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1-u^{2}=0", "result": "-u^{2}=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1-u^{2}-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "-u^{2}=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-u^{2}=-1", "result": "u^{2}=1", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-u^{2}}{-1}=\\frac{-1}{-1}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=1" } ], "meta": { "interimType": "Divide Both Sides Specific 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{1},\\:u=-\\sqrt{1}" }, { "type": "interim", "title": "$$\\sqrt{1}=1$$", "input": "\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KfzlHGGU7KN8vfEO0eL8NN13jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fScubCnYZOJ5L8/2gsdymw1PSOscTE6qsKVI9GkIdY/eI=" } }, { "type": "interim", "title": "$$-\\sqrt{1}=-1$$", "input": "-\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kWDE1Jsjy5jGSP2mctwcnCAn9lkDfZkicUGkO3EF+IpIQKToZa7Vmz9RWrIHzooCMHIu6EZfZrJ7HpyNTqg74lPlyk515FWfACaTxs0eUEM=" } }, { "type": "step", "result": "u=1,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=1,\\:\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=1{\\quad:\\quad}x=2πn$$", "input": "\\cos\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=1$$", "result": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=0+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-1{\\quad:\\quad}x=π+2πn$$", "input": "\\cos\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=-1$$", "result": "x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn" }, { "type": "step", "primary": "Since the equation is undefined for:$${\\quad}2πn,\\:π+2πn$$", "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": "\\frac{(1+\\cot^{2}(x))}{\\cos^{2}(x)}-\\cot^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }