2sin^2(x/2)+cos(x)=1+sin(x)

{ "query": { "display": "$$2\\sin^{2}\\left(\\frac{x}{2}\\right)+\\cos\\left(x\\right)=1+\\sin\\left(x\\right)$$", "symbolab_question": "EQUATION#2\\sin^{2}(\\frac{x}{2})+\\cos(x)=1+\\sin(x)" }, "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": "$$2\\sin^{2}\\left(\\frac{x}{2}\\right)+\\cos\\left(x\\right)=1+\\sin\\left(x\\right){\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "2\\sin^{2}\\left(\\frac{x}{2}\\right)+\\cos\\left(x\\right)=1+\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$1+\\sin\\left(x\\right)$$ from both sides", "result": "2\\sin^{2}\\left(\\frac{x}{2}\\right)+\\cos\\left(x\\right)-1-\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos\\left(x\\right)-\\sin\\left(x\\right)+2\\sin^{2}\\left(\\frac{x}{2}\\right)", "result": "-\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$1-2\\sin^{2}\\left(x\\right)=\\cos\\left(2x\\right)$$", "secondary": [ "$$-1+2\\sin^{2}\\left(x\\right)=-\\cos\\left(2x\\right)$$" ], "result": "=\\cos\\left(x\\right)-\\sin\\left(x\\right)-\\cos\\left(2\\cdot\\:\\frac{x}{2}\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(x\\right)-\\sin\\left(x\\right)-\\cos\\left(2\\cdot\\:\\frac{x}{2}\\right):{\\quad}-\\sin\\left(x\\right)$$", "input": "\\cos\\left(x\\right)-\\sin\\left(x\\right)-\\cos\\left(2\\cdot\\:\\frac{x}{2}\\right)", "result": "=-\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{x}{2}\\::{\\quad}x$$", "input": "2\\cdot\\:\\frac{x}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=x" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\cos\\left(x\\right)-\\sin\\left(x\\right)-\\cos\\left(x\\right)" }, { "type": "step", "primary": "Add similar elements: $$\\cos\\left(x\\right)-\\cos\\left(x\\right)=0$$", "result": "=-\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iE7XUImDsKaPOXjf7SwGzoCCwnrG6YHAZoSmWhSWWsrE9mX+nskMceR3nwiAVW8sLTrWWMFI8l4Q07DZ5+hJayWe7O5b9aX/U5M/Rbl6+FVwXkwSgKeU7zr2aSFYDfnjixpjZyz1hn/Tx95aZo6CSIs6D9JqHi4yUTdHoabLXe30YEGAYUWD6U18e8QBU3Iamkbnybg9aS9gUhfowYe7rkuIXgzNu3qzDJJYVVN57iY=" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VLGVgz+X+emXvMLh5jncmppB1a38tWXv3iJQ/91st215SmWG5AOdXc2mV99OwqKyGfMEMmdga/GoG2PKpC1Lelj5tLbu6quyLq/4/G1EXhBDquSOCisf6+ubu5E0rz8qbAhCvdyklUsQ/Veq+q5znnRkS3dlcCKpQTQcheuut7MkmLRx2SCfDYc+NFWlwczFlshyJhXONoCiURo2RU420EFGmlM2AwWiez1syRPMgeig" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-\\sin\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-\\sin\\left(x\\right)}{-1}=\\frac{0}{-1}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "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": "2\\sin^{2}(\\frac{x}{2})+\\cos(x)-1-\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }