(cos^{2022}(x))/((sin^{2022)(x)+cos^{2022}(x))}=1

{ "query": { "display": "$$\\frac{\\cos^{2022}\\left(x\\right)}{\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}=1$$", "symbolab_question": "EQUATION#\\frac{\\cos^{2022}(x)}{(\\sin^{2022}(x)+\\cos^{2022}(x))}=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\cos^{2022}\\left(x\\right)}{\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}=1{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\frac{\\cos^{2022}\\left(x\\right)}{\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}=1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\frac{\\cos^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}-1=0" }, { "type": "interim", "title": "Simplify $$\\frac{\\cos^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}-1:{\\quad}-\\frac{\\sin^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}$$", "input": "\\frac{\\cos^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}-1", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}$$", "result": "=\\frac{\\cos^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}-\\frac{1\\cdot\\:\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\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{\\cos^{2022}\\left(x\\right)-1\\cdot\\:\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)=\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)$$", "result": "=\\frac{\\cos^{2022}\\left(x\\right)-\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}" }, { "type": "interim", "title": "Expand $$\\cos^{2022}\\left(x\\right)-\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right):{\\quad}-\\sin^{2022}\\left(x\\right)$$", "input": "\\cos^{2022}\\left(x\\right)-\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)", "result": "=\\frac{-\\sin^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$-\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right):{\\quad}-\\sin^{2022}\\left(x\\right)-\\cos^{2022}\\left(x\\right)$$", "input": "-\\left(\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)\\right)", "result": "=\\cos^{2022}\\left(x\\right)-\\sin^{2022}\\left(x\\right)-\\cos^{2022}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\sin^{2022}\\left(x\\right)\\right)-\\left(\\cos^{2022}\\left(x\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-\\sin^{2022}\\left(x\\right)-\\cos^{2022}\\left(x\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Add similar elements: $$\\cos^{2022}\\left(x\\right)-\\cos^{2022}\\left(x\\right)=0$$", "result": "=-\\sin^{2022}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eMAmCKpiippspwI9QFezMjHAYreoA8X2BD+Quudr4aLY10HjFwEQEg2N6h4ARtl6Q8pkGcgtlPCK62pgtZcgMKORWLXkjysF58uLgjK3bCvP5dQILn7sNBnJkpr/qmcoP8vQyhiD4JSfqjIvcQ7tiqvhme4+pJbvIQ3GyaucLUE5wib2SILS+lfHTTLJvhcNa3DPkvzFuUQbgfXxPtOcR4AF47coyCH0GKtjfcKJefC2ZGLQTd4LQd7/ERKlQuFW" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\sin^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "-\\frac{\\sin^{2022}\\left(x\\right)}{\\sin^{2022}\\left(x\\right)+\\cos^{2022}\\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": "-\\left(\\sin^{2022}\\left(x\\right)\\right)=0" }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-\\left(\\sin^{2022}\\left(x\\right)\\right)=0", "result": "\\sin^{2022}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-\\sin^{2022}\\left(x\\right)}{-1}=\\frac{0}{-1}" }, { "type": "step", "primary": "Simplify", "result": "\\sin^{2022}\\left(x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zSQREjT9ODtZxHlFLK1q98EpkrWBY96re9bJeX4lZZt6plOa0LSfM3baidJI4aQpgIoSPGKTmu2I3GaK7xAsLMWXIUHnK302cSzWmTJ9f+er8iUxc76TdUZEujsTpIokojV+8f6LWMH6eQftnraPow7upzez8Ly2EObkVpSZ9PxNz4bOy87AdO8OBNpVeSyBkM6UjUQPgG0w1jOVDhYEdYRaMe7VuinvQ0IJdEqmbZ9H30koNqZdcC0ISYHUsNsGlsMxPN+Du666aYYZH5j60ZIcOPtBjEwf5NB6rJgRkL46ykpGEGxsjZM9XQCCmA2ItQunUuCLvpwGYAhvio+mTW9a4n83XX1fDim1PknhUaB6pfF1z6umzUJTJvt+ojYZc7mh9SH8T2tyRGdJc3t6bSG3bx2f4oU+KqnyAA9f6CI=" } }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "\\sin\\left(x\\right)=0" }, { "type": "step" }, { "type": "step", "primary": "The solution is", "result": "\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=π+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=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 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,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{\\cos^{2022}(x)}{(\\sin^{2022}(x)+\\cos^{2022}(x))}-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }