cos(2x)-sin(2x)=1

{ "query": { "display": "$$\\cos\\left(2x\\right)-\\sin\\left(2x\\right)=1$$", "symbolab_question": "EQUATION#\\cos(2x)-\\sin(2x)=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn,x=\\frac{3π}{4}+πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n,x=135^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(2x\\right)-\\sin\\left(2x\\right)=1{\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{3π}{4}+πn$$", "input": "\\cos\\left(2x\\right)-\\sin\\left(2x\\right)=1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\cos\\left(2x\\right)-\\sin\\left(2x\\right)-1=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos\\left(2x\\right)-\\sin\\left(2x\\right)", "result": "-2\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=-1+\\cos\\left(2x\\right)-2\\sin\\left(x\\right)\\cos\\left(x\\right)" }, { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=1-2\\sin^{2}\\left(x\\right)$$", "result": "=-1+1-2\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)\\sin\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-2\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)\\sin\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VL7D3CKrq0jBf68C0F/oVeYaM37KF/jK0q16Pg+4JBw35yFOVAzu8w4itJwsMopep5nX/LTRLwEhE6Awmk/6zhUIAqZBkW8uMb72Rzx6Zt+t4vBTTenjlx5r6wcPYjU8rsEyH9PpAjyt5Wo2WmM2ruZkS3dlcCKpQTQcheuut7MkmLRx2SCfDYc+NFWlwczFlshyJhXONoCiURo2RU420EFGmlM2AwWiez1syRPMgeig" } }, { "type": "interim", "title": "Factor $$-2\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)\\sin\\left(x\\right):{\\quad}-2\\sin\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)$$", "input": "-2\\sin^{2}\\left(x\\right)-2\\cos\\left(x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\sin^{2}\\left(x\\right)=\\sin\\left(x\\right)\\sin\\left(x\\right)$$" ], "result": "=-2\\sin\\left(x\\right)\\sin\\left(x\\right)-2\\sin\\left(x\\right)\\cos\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$-2\\sin\\left(x\\right)$$", "result": "=-2\\sin\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "-2\\sin\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(x\\right)=0\\lor\\:\\sin\\left(x\\right)+\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin\\left(x\\right)=0", "steps": [ { "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", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)+\\cos\\left(x\\right)=0{\\quad:\\quad}x=\\frac{3π}{4}+πn$$", "input": "\\sin\\left(x\\right)+\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(x\\right)+\\cos\\left(x\\right)=0", "result": "\\tan\\left(x\\right)+1=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\cos\\left(x\\right),\\:\\cos\\left(x\\right)\\neq0$$", "result": "\\frac{\\sin\\left(x\\right)+\\cos\\left(x\\right)}{\\cos\\left(x\\right)}=\\frac{0}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}+1=0" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\tan\\left(x\\right)$$", "result": "\\tan\\left(x\\right)+1=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFAyMsO/aAtw3YZD7XkUL8T4FMS0G0HgjUXwc3RQOmn9BYznLdIBaHD4ibBnEILJTDXUHW1mE5UQUTIwuQd6ag+zflewpXsN2Z5/tCbT2a61ddL6nfREQPpDlvjUlVNvdtU3kCh3oevUunZ7/b0qFKBTCPNlUYj52tc4Y8B80I30B5l60FyrIpKpS8qSplgYxPCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\tan\\left(x\\right)+1=0", "result": "\\tan\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\tan\\left(x\\right)+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\tan\\left(x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{4}+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=\\frac{3π}{4}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{3π}{4}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(2x)-\\sin(2x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }