6tan(x)(1-cos^2(x))=tan(x)

{ "query": { "display": "$$6\\tan\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)=\\tan\\left(x\\right)$$", "symbolab_question": "EQUATION#6\\tan(x)(1-\\cos^{2}(x))=\\tan(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=πn,x=2.72105…+2πn,x=-2.72105…+2πn,x=0.42053…+2πn,x=2π-0.42053…+2πn", "degrees": "x=0^{\\circ }+180^{\\circ }n,x=155.90515…^{\\circ }+360^{\\circ }n,x=-155.90515…^{\\circ }+360^{\\circ }n,x=24.09484…^{\\circ }+360^{\\circ }n,x=335.90515…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$6\\tan\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)=\\tan\\left(x\\right){\\quad:\\quad}x=πn,\\:x=2.72105…+2πn,\\:x=-2.72105…+2πn,\\:x=0.42053…+2πn,\\:x=2π-0.42053…+2πn$$", "input": "6\\tan\\left(x\\right)\\left(1-\\cos^{2}\\left(x\\right)\\right)=\\tan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\tan\\left(x\\right)$$ from both sides", "result": "5\\tan\\left(x\\right)-6\\cos^{2}\\left(x\\right)\\tan\\left(x\\right)=0" }, { "type": "interim", "title": "Factor $$5\\tan\\left(x\\right)-6\\cos^{2}\\left(x\\right)\\tan\\left(x\\right):{\\quad}-\\tan\\left(x\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}\\right)$$", "input": "5\\tan\\left(x\\right)-6\\cos^{2}\\left(x\\right)\\tan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$-\\tan\\left(x\\right)$$", "result": "=-\\tan\\left(x\\right)\\left(-5+6\\cos^{2}\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$6\\cos^{2}\\left(x\\right)-5:{\\quad}\\left(\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}\\right)$$", "input": "6\\cos^{2}\\left(x\\right)-5", "steps": [ { "type": "interim", "title": "Rewrite $$6\\cos^{2}\\left(x\\right)-5$$ as $$\\left(\\sqrt{6}\\cos\\left(x\\right)\\right)^{2}-\\left(\\sqrt{5}\\right)^{2}$$", "input": "6\\cos^{2}\\left(x\\right)-5", "result": "=\\left(\\sqrt{6}\\cos\\left(x\\right)\\right)^{2}-\\left(\\sqrt{5}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$6=\\left(\\sqrt{6}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{6}\\right)^{2}\\cos^{2}\\left(x\\right)-5", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$5=\\left(\\sqrt{5}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{6}\\right)^{2}\\cos^{2}\\left(x\\right)-\\left(\\sqrt{5}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\left(\\sqrt{6}\\right)^{2}\\cos^{2}\\left(x\\right)=\\left(\\sqrt{6}\\cos\\left(x\\right)\\right)^{2}$$" ], "result": "=\\left(\\sqrt{6}\\cos\\left(x\\right)\\right)^{2}-\\left(\\sqrt{5}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(\\sqrt{6}\\cos\\left(x\\right)\\right)^{2}-\\left(\\sqrt{5}\\right)^{2}=\\left(\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}\\right)$$" ], "result": "=\\left(\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\tan\\left(x\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "-\\tan\\left(x\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}\\right)\\left(\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\tan\\left(x\\right)=0\\lor\\:\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}=0\\lor\\:\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}=0" }, { "type": "interim", "title": "$$\\tan\\left(x\\right)=0{\\quad:\\quad}x=πn$$", "input": "\\tan\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=0$$", "result": "x=0+πn", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=0+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+πn:{\\quad}x=πn$$", "input": "x=0+πn", "steps": [ { "type": "step", "primary": "$$0+πn=πn$$", "result": "x=πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}=0{\\quad:\\quad}x=\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn$$", "input": "\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{5}\\:$$to the right side", "input": "\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}=0", "result": "\\sqrt{6}\\cos\\left(x\\right)=-\\sqrt{5}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{5}$$ from both sides", "result": "\\sqrt{6}\\cos\\left(x\\right)+\\sqrt{5}-\\sqrt{5}=0-\\sqrt{5}" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{6}\\cos\\left(x\\right)=-\\sqrt{5}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{6}$$", "input": "\\sqrt{6}\\cos\\left(x\\right)=-\\sqrt{5}", "result": "\\cos\\left(x\\right)=-\\sqrt{\\frac{5}{6}}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{6}$$", "result": "\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}=\\frac{-\\sqrt{5}}{\\sqrt{6}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}=\\frac{-\\sqrt{5}}{\\sqrt{6}}", "result": "\\cos\\left(x\\right)=-\\sqrt{\\frac{5}{6}}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}:{\\quad}\\cos\\left(x\\right)$$", "input": "\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{6}$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMZKiLtd/z2LfGuMvD48rjnUMkx53+40dilvAmZIYRSTDq47vuWedXv2WUg94ER8IwasnM+lFdK+lbOA6QVAeavaBBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2UI4TKsSSybUmGnaZPB9aJtyuldobPmLQIz5Xi1h26RkFkVDw5yEF2DEw4msQzKTew==" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\sqrt{5}}{\\sqrt{6}}:{\\quad}-\\sqrt{\\frac{5}{6}}$$", "input": "\\frac{-\\sqrt{5}}{\\sqrt{6}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\sqrt{5}}{\\sqrt{6}}" }, { "type": "step", "primary": "Combine same powers : $$\\frac{\\sqrt{x}}{\\sqrt{y}}=\\sqrt{\\frac{x}{y}}$$", "result": "=-\\sqrt{\\frac{5}{6}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70c9DTe1ubFlIcViNwxWNeco+iBPnJW3YGbbpjEYlb24tOtZYwUjyXhDTsNnn6ElrJZ7s7lv1pf9Tkz9FuXr4VZ0tj/9ppoVpu+bsRwNWhGD/P/+v51eTuH2/F4MIu5mqcoXe7Y4hvhsUDcQ7xpGVJf/u7H7DbOypDnKV6g7eetjkIhvakUt4IE7NCpNWfZTWialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "\\cos\\left(x\\right)=-\\sqrt{\\frac{5}{6}}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=-\\sqrt{\\frac{5}{6}}", "result": "x=\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=-\\sqrt{\\frac{5}{6}}$$", "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(-\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}=0{\\quad:\\quad}x=\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn$$", "input": "\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{5}\\:$$to the right side", "input": "\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}=0", "result": "\\sqrt{6}\\cos\\left(x\\right)=\\sqrt{5}", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{5}$$ to both sides", "result": "\\sqrt{6}\\cos\\left(x\\right)-\\sqrt{5}+\\sqrt{5}=0+\\sqrt{5}" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{6}\\cos\\left(x\\right)=\\sqrt{5}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ECl81GL+sHp+UX1O7owT7b4CT66by0jKZIwCuuTNHhjuxJbcOYXqXvwi88hfZHpfhkxIvVgVw839WgeD4S9D3xA5YFRug4J+fK4IBlJRmXki/ekth8rJBsoKyGGRpkdRFA2vHp/GlMJljHZhzfwfOShPS5ji8EItibtV+jZARuafdxWeCOIxvlQy1l0H0S3e0hYzF6vxjSZoMSInCY/PPQbO95/2yiZOrh8Vdx/OHPuatTLMy0jFYlw9LG2LLE8qKt3K064XvCieQ7wILBhewCwfY4N34mQKtyk5NuZIlqEJaotGU4CQeFeDXjgYDaDbJF7Y9kZd0P63mq6FYn/mRtWBqIfAxPCuGoyiHyWjngi/37j2cKu7zNO0u0Sr7TfuYmNXJn93OP6ij2GPtA34otxaaiLjd2q2Hka3ZyPExBeS97wBhDJOMHA38nLxSVPqtwQOIpzTgKlHvB9zatRF0gwKAu7Z3tN/RgLPjlzyRw8i5OoNNprqQV2UgQUGESqFlvjhqQs5XJJkHv2GQUE6glcrAOsROfiVgP+wtWyh0afthvwKEoR3ksHNHbCkrb6kcwDIfxbr/Pq2vWjsXBdpmP+Gc7uNDn5rFMiDyv69niKS9nJYJSn323kBOgLLJJSH3CPQJK0v1YFQ8gsqE/wU78R9H9EnWAR76jj5XngDhpjzGL3BXreZc5/AN9v+azZd" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{6}$$", "input": "\\sqrt{6}\\cos\\left(x\\right)=\\sqrt{5}", "result": "\\cos\\left(x\\right)=\\sqrt{\\frac{5}{6}}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{6}$$", "result": "\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}=\\frac{\\sqrt{5}}{\\sqrt{6}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}=\\frac{\\sqrt{5}}{\\sqrt{6}}", "result": "\\cos\\left(x\\right)=\\sqrt{\\frac{5}{6}}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}:{\\quad}\\cos\\left(x\\right)$$", "input": "\\frac{\\sqrt{6}\\cos\\left(x\\right)}{\\sqrt{6}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{6}$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMZKiLtd/z2LfGuMvD48rjnUMkx53+40dilvAmZIYRSTDq47vuWedXv2WUg94ER8IwasnM+lFdK+lbOA6QVAeavaBBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2UI4TKsSSybUmGnaZPB9aJtyuldobPmLQIz5Xi1h26RkFkVDw5yEF2DEw4msQzKTew==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{5}}{\\sqrt{6}}:{\\quad}\\sqrt{\\frac{5}{6}}$$", "input": "\\frac{\\sqrt{5}}{\\sqrt{6}}", "steps": [ { "type": "step", "primary": "Combine same powers : $$\\frac{\\sqrt{x}}{\\sqrt{y}}=\\sqrt{\\frac{x}{y}}$$", "result": "=\\sqrt{\\frac{5}{6}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMTjeh/UWaANBiQIuMaFkuDtV00rpv8+ZC6TM10tVCSHsoDcnz6I6EbFIZGDoyqUAxyvmHggofEc9y3nQOEIjQGBkS3dlcCKpQTQcheuut7MkdkqXaZXgCXO2IClMAWKDPeb/3YM6fh1Iwg78vHbkfjMUBnAGUajMnfZKIw5RGKgSJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "\\cos\\left(x\\right)=\\sqrt{\\frac{5}{6}}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=\\sqrt{\\frac{5}{6}}", "result": "x=\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=\\sqrt{\\frac{5}{6}}$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=2π-\\arccos\\left(a\\right)+2πn$$" ], "result": "x=\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=πn,\\:x=\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=-\\arccos\\left(-\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt{\\frac{5}{6}}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=πn,\\:x=2.72105…+2πn,\\:x=-2.72105…+2πn,\\:x=0.42053…+2πn,\\:x=2π-0.42053…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "6\\tan(x)(1-\\cos^{2}(x))-\\tan(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }