sec(x)cot(x)=-2

{ "query": { "display": "$$\\sec\\left(x\\right)\\cot\\left(x\\right)=-2$$", "symbolab_question": "EQUATION#\\sec(x)\\cot(x)=-2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{7π}{6}+2πn,x=\\frac{11π}{6}+2πn", "degrees": "x=210^{\\circ }+360^{\\circ }n,x=330^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sec\\left(x\\right)\\cot\\left(x\\right)=-2{\\quad:\\quad}x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "\\sec\\left(x\\right)\\cot\\left(x\\right)=-2", "steps": [ { "type": "step", "primary": "Subtract $$-2$$ from both sides", "result": "\\sec\\left(x\\right)\\cot\\left(x\\right)+2=0" }, { "type": "interim", "title": "Express with sin, cos", "input": "2+\\cot\\left(x\\right)\\sec\\left(x\\right)", "result": "\\frac{1+2\\sin\\left(x\\right)}{\\sin\\left(x\\right)}=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\cot\\left(x\\right)=\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=2+\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}\\sec\\left(x\\right)" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\sec\\left(x\\right)=\\frac{1}{\\cos\\left(x\\right)}$$", "result": "=2+\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$2+\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}:{\\quad}\\frac{2\\sin\\left(x\\right)+1}{\\sin\\left(x\\right)}$$", "input": "2+\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}", "result": "=\\frac{2\\sin\\left(x\\right)+1}{\\sin\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}=\\frac{1}{\\sin\\left(x\\right)}$$", "input": "\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\cos\\left(x\\right)\\cdot\\:1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\cos\\left(x\\right)$$", "result": "=\\frac{1}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Kqg/3+OyCpVPWw23luNHAoYQB6PL2t1fNTT4ASxE0IsZIpiGaO8VDd3tueArJZWfvgODWgpJNtyK6jgKaIhT2KuO77lnnV79llIPeBEfCMG6/Fpfs2IISc5iSynSoG6XFYdDNkfoQHH3zJ0XLKuAKkkaXvzSgca9NJ3DtVeGw6WgFrjcswyY9dzktAEhWQmDeJePQinV2hdE8gkJAU67ZN46/3S4BGHnCGL+vPq29KWk78YeyxmZOwHOdxpcDVDTJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2+\\frac{1}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2\\sin\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=\\frac{2\\sin\\left(x\\right)}{\\sin\\left(x\\right)}+\\frac{1}{\\sin\\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{2\\sin\\left(x\\right)+1}{\\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+2\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1+2\\sin\\left(x\\right)=0", "result": "2\\sin\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1+2\\sin\\left(x\\right)-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "2\\sin\\left(x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\sin\\left(x\\right)=-1", "result": "\\sin\\left(x\\right)=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\sin\\left(x\\right)}{2}=\\frac{-1}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(x\\right)=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-\\frac{1}{2}$$", "result": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sec(x)\\cot(x)+2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }