(tan(x)+cot(x))/(csc(x))=cos(x)

{ "query": { "display": "$$\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}=\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#\\frac{\\tan(x)+\\cot(x)}{\\csc(x)}=\\cos(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": "$$\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}=\\cos\\left(x\\right){\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}=\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos\\left(x\\right)$$ from both sides", "result": "\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}-\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "Simplify $$\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}-\\cos\\left(x\\right):{\\quad}\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)-\\cos\\left(x\\right)\\csc\\left(x\\right)}{\\csc\\left(x\\right)}$$", "input": "\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}-\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\cos\\left(x\\right)=\\frac{\\cos\\left(x\\right)\\csc\\left(x\\right)}{\\csc\\left(x\\right)}$$", "result": "=\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)}{\\csc\\left(x\\right)}-\\frac{\\cos\\left(x\\right)\\csc\\left(x\\right)}{\\csc\\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{\\tan\\left(x\\right)+\\cot\\left(x\\right)-\\cos\\left(x\\right)\\csc\\left(x\\right)}{\\csc\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{\\tan\\left(x\\right)+\\cot\\left(x\\right)-\\cos\\left(x\\right)\\csc\\left(x\\right)}{\\csc\\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": "\\tan\\left(x\\right)+\\cot\\left(x\\right)-\\cos\\left(x\\right)\\csc\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cot\\left(x\\right)+\\tan\\left(x\\right)-\\cos\\left(x\\right)\\csc\\left(x\\right)", "result": "\\tan\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "$$\\csc\\left(x\\right)\\cos\\left(x\\right)=\\cot\\left(x\\right)$$", "input": "\\csc\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "Express with sin, cos", "input": "\\csc\\left(x\\right)\\cos\\left(x\\right)", "result": "=\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "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)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sin\\left(x\\right)}\\cos\\left(x\\right):{\\quad}\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\frac{1}{\\sin\\left(x\\right)}\\cos\\left(x\\right)", "result": "=\\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{1\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(x\\right)=\\cos\\left(x\\right)$$", "result": "=\\frac{\\cos\\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": "Use the basic trigonometric identity: $$\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}=\\cot\\left(x\\right)$$", "result": "=\\cot\\left(x\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\cot\\left(x\\right)+\\tan\\left(x\\right)-\\cot\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\tan\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Frwf5jMMuALOhpxPA+cZVgTQv6Bhz6IR4Cj+8+Bbj3VrNYQrXitounAin1Yzcy5VsrkXJxYJ4fAIo4WDY4askW8mWMl+hBjAHADIIzgA6jy2oyMT9jq45AQhAkHfGusO3JjQAKeLTk4dvGjGcmccYObxHHqBGPll5mcoxZeXMAgYcvEp+ArJ7IgQQisySYDH4va4o6ZZQt6EK5Zci1d56K6g7caZnrJReWEr4vU9em8=" } }, { "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" }, { "type": "step", "primary": "Since the equation is undefined for:$${\\quad}πn$$", "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{\\tan(x)+\\cot(x)}{\\csc(x)}-\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }