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

{ "query": { "display": "$$\\left(1-\\sin\\left(x\\right)\\right)\\left(1-\\sin\\left(x\\right)\\right)=\\cos^{2}\\left(x\\right)$$", "symbolab_question": "EQUATION#(1-\\sin(x))(1-\\sin(x))=\\cos^{2}(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn,x=2πn,x=π+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n,x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(1-\\sin\\left(x\\right)\\right)\\left(1-\\sin\\left(x\\right)\\right)=\\cos^{2}\\left(x\\right){\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=2πn,\\:x=π+2πn$$", "input": "\\left(1-\\sin\\left(x\\right)\\right)\\left(1-\\sin\\left(x\\right)\\right)=\\cos^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos^{2}\\left(x\\right)$$ from both sides", "result": "\\left(1-\\sin\\left(x\\right)\\right)^{2}-\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\left(1-\\sin\\left(x\\right)\\right)^{2}-\\cos^{2}\\left(x\\right)", "result": "-2\\sin\\left(x\\right)+2\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$" ], "result": "=\\left(1-\\sin\\left(x\\right)\\right)^{2}-\\left(1-\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\left(1-\\sin\\left(x\\right)\\right)^{2}-\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)$$", "input": "\\left(1-\\sin\\left(x\\right)\\right)^{2}-\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\left(1-\\sin\\left(x\\right)\\right)^{2}:{\\quad}1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "result": "=1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)-\\left(1-\\sin^{2}\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=1,\\:\\:b=\\sin\\left(x\\right)$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=1^{2}-2\\cdot\\:1\\cdot\\:\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$1^{2}-2\\cdot\\:1\\cdot\\:\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right):{\\quad}1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "input": "1^{2}-2\\cdot\\:1\\cdot\\:\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)", "result": "=1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=1-2\\cdot\\:1\\cdot\\:\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}-1+\\sin^{2}\\left(x\\right)$$", "input": "-\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)-1+\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(1\\right)-\\left(-\\sin^{2}\\left(x\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-1+\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Simplify $$1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)-1+\\sin^{2}\\left(x\\right):{\\quad}2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)$$", "input": "1-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)-1+\\sin^{2}\\left(x\\right)", "result": "=2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-2\\sin\\left(x\\right)+\\sin^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)+1-1" }, { "type": "step", "primary": "Add similar elements: $$\\sin^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=2\\sin^{2}\\left(x\\right)$$", "result": "=-2\\sin\\left(x\\right)+2\\sin^{2}\\left(x\\right)+1-1" }, { "type": "step", "primary": "$$1-1=0$$", "result": "=2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ahdJAxDQBq1Bkmmv/nUb16XHu9DsuMDUib4lfg22qYtDymQZyC2U8IrramC1lyAwo5FYteSPKwXny4uCMrdsK4kUOW7x0OLtN8wUSNZo/3MPdnKco50MXmBViyyUVwKsTeQKHeh69S6dnv9vSoUoFMoLafipPAdpMgO38j+yqZMTddLftv+kM2UzejGHKyEeJ+3nocXA07Wfg7SM4Nvc1Q==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xNIUpMDzZ16UzMZLPhf6ZCD8nT05eDO3q/lSa47nemFBjGZ+c+wIWHxP5VQdJDji8wQyZ2Br8agbY8qkLUt6WPm0tu7qq7Iur/j8bUReEEOq5I4KKx/r65u7kTSvPyps1jrJRGG8ozExA+YH/4DUb34Ic+4AhJ5g9FmIFHx96sNFKk3fejFkyiOiq9iG9IkAO/B7UtNBfw19bDRpX2AlzDEZOhzOLFFCWOVe0TAs9Ps/tVtg7k21KzJIFYRgfW7H" } }, { "type": "interim", "title": "Solve by substitution", "input": "-2\\sin\\left(x\\right)+2\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=1,\\:\\sin\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "-2u+2u^{2}=0" }, { "type": "interim", "title": "$$-2u+2u^{2}=0{\\quad:\\quad}u=1,\\:u=0$$", "input": "-2u+2u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "2u^{2}-2u=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "2u^{2}-2u=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\cdot\\:0}}{2\\cdot\\:2}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=2,\\:b=-2,\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\cdot\\:0}}{2\\cdot\\:2}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\cdot\\:0}=2$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:2\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:2}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}-4\\cdot\\:2\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{2^{2}-0}" }, { "type": "step", "primary": "$$2^{2}-0=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTjQqsHVe47/y2heJhgcMHnSGcE76jDOroaIH3gr3HS6TzMFYmi1F5Hg/ibpEToVnYwSc/te/iOpB2yBHqsIcuJnPYvWlDOvQwlh4TEmetdSSNCqwdV7jv/LaF4mGBwwedFXNpVIlvpgHOi/3rrWD6As=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-2\\right)+2}{2\\cdot\\:2},\\:{u}_{2}=\\frac{-\\left(-2\\right)-2}{2\\cdot\\:2}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)+2}{2\\cdot\\:2}:{\\quad}1$$", "input": "\\frac{-\\left(-2\\right)+2}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+2}{2\\cdot\\:2}" }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\frac{4}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Fv9gVNWyI/X8TvzzSZV8n5Ket2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiK7J5E5gGi2xwchkRMjoVJ7ir6EdYdh/n2c4DPMiuUGOJhVjXCvkuV0dlNfJSARrUYx0ij68Lvr+79BVqXMQAj8w==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)-2}{2\\cdot\\:2}:{\\quad}0$$", "input": "\\frac{-\\left(-2\\right)-2}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-2}{2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$2-2=0$$", "result": "=\\frac{0}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{0}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzZ+Rg5UPtRAM2q6UZuzBpKet2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09ir6EdYdh/n2c4DPMiuUGOIEhkNIoEcCE5KusvGzEjW2x0ij68Lvr+79BVqXMQAj8w==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=1,\\:u=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=1,\\:\\sin\\left(x\\right)=0" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=1{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=1$$", "result": "x=\\frac{π}{2}+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{π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "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": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn,\\: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": "(1-\\sin(x))(1-\\sin(x))-\\cos^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }