cos(2x)csc^2(x)=2cos(2x)

{ "query": { "display": "$$\\cos\\left(2x\\right)\\csc^{2}\\left(x\\right)=2\\cos\\left(2x\\right)$$", "symbolab_question": "EQUATION#\\cos(2x)\\csc^{2}(x)=2\\cos(2x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{4}+πn,x=\\frac{3π}{4}+πn", "degrees": "x=45^{\\circ }+180^{\\circ }n,x=135^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(2x\\right)\\csc^{2}\\left(x\\right)=2\\cos\\left(2x\\right){\\quad:\\quad}x=\\frac{π}{4}+πn,\\:x=\\frac{3π}{4}+πn$$", "input": "\\cos\\left(2x\\right)\\csc^{2}\\left(x\\right)=2\\cos\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$2\\cos\\left(2x\\right)$$ from both sides", "result": "\\csc^{2}\\left(x\\right)\\cos\\left(2x\\right)-2\\cos\\left(2x\\right)=0" }, { "type": "interim", "title": "Factor $$\\csc^{2}\\left(x\\right)\\cos\\left(2x\\right)-2\\cos\\left(2x\\right):{\\quad}\\cos\\left(2x\\right)\\left(\\csc\\left(x\\right)+\\sqrt{2}\\right)\\left(\\csc\\left(x\\right)-\\sqrt{2}\\right)$$", "input": "\\csc^{2}\\left(x\\right)\\cos\\left(2x\\right)-2\\cos\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$\\cos\\left(2x\\right)$$", "result": "=\\cos\\left(2x\\right)\\left(\\csc^{2}\\left(x\\right)-2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$\\csc^{2}\\left(x\\right)-2:{\\quad}\\left(\\csc\\left(x\\right)+\\sqrt{2}\\right)\\left(\\csc\\left(x\\right)-\\sqrt{2}\\right)$$", "input": "\\csc^{2}\\left(x\\right)-2", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$2=\\left(\\sqrt{2}\\right)^{2}$$" ], "result": "=\\csc^{2}\\left(x\\right)-\\left(\\sqrt{2}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\csc^{2}\\left(x\\right)-\\left(\\sqrt{2}\\right)^{2}=\\left(\\csc\\left(x\\right)+\\sqrt{2}\\right)\\left(\\csc\\left(x\\right)-\\sqrt{2}\\right)$$" ], "result": "=\\left(\\csc\\left(x\\right)+\\sqrt{2}\\right)\\left(\\csc\\left(x\\right)-\\sqrt{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\cos\\left(2x\\right)\\left(\\csc\\left(x\\right)+\\sqrt{2}\\right)\\left(\\csc\\left(x\\right)-\\sqrt{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\cos\\left(2x\\right)\\left(\\csc\\left(x\\right)+\\sqrt{2}\\right)\\left(\\csc\\left(x\\right)-\\sqrt{2}\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\cos\\left(2x\\right)=0\\lor\\:\\csc\\left(x\\right)+\\sqrt{2}=0\\lor\\:\\csc\\left(x\\right)-\\sqrt{2}=0" }, { "type": "interim", "title": "$$\\cos\\left(2x\\right)=0{\\quad:\\quad}x=\\frac{π}{4}+πn,\\:x=\\frac{3π}{4}+πn$$", "input": "\\cos\\left(2x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(2x\\right)=0$$", "result": "2x=\\frac{π}{2}+2πn,\\:2x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2x=\\frac{π}{2}+2πn,\\:2x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{π}{2}+2πn:{\\quad}x=\\frac{π}{4}+πn$$", "input": "2x=\\frac{π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{π}{2}+2πn", "result": "x=\\frac{π}{4}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{π}{2}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{π}{2}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{π}{4}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{2}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{π}{4}+πn$$", "input": "\\frac{\\frac{π}{2}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{2}}{2}=\\frac{π}{4}$$", "input": "\\frac{\\frac{π}{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajalBIoKqdyCxArWzxWIW6vbNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G1O1T0snFOPqKXL+S6MxVmm+/XUfFY44oQy7yrU6sldlPOBDgzHOCMljoJao6lBoOl+VlAtW4e5GbXcVxB3X2iQ=" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{π}{4}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajYuMoFzG5l2Co8OYeWnq/n3Cl1oDQxTLS01lmG4qDSnCzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhuq35cfBN7Lz4zKHbnarYjveqXxdc+rps1CUyb7fqI2GWOC1RGw1nyYXaraVAOvzodKeHC05xpDDzQIDq9pLXcZMraKqGjoVZ5Ymz5GNk+xPPotFMd5aa71+dAhX5ea6wE=" } }, { "type": "step", "result": "x=\\frac{π}{4}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{3π}{2}+2πn:{\\quad}x=\\frac{3π}{4}+πn$$", "input": "2x=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{3π}{2}+2πn", "result": "x=\\frac{3π}{4}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{3π}{4}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{3π}{4}+πn$$", "input": "\\frac{\\frac{3π}{2}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{2}}{2}=\\frac{3π}{4}$$", "input": "\\frac{\\frac{3π}{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{3π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZJTpMqJaq9bfSnb9kPhskfdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldREkIgLCzuVxsE8bZJAi+zLMWtOub2dAxgaWacwKjDMkBc6krFpzteeOSStKZ2st+3LSXRhuEYkItk0ZmqvQzatokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{3π}{4}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZ5yKgO88C1F1xZtWMQHGNW2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXa3zET8py0nz91WHtTkJXUTV9uDRRJh/E3msxuJMsId28LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJcIdcrEX3CFtz7dlmgYCS4trsGj7x6oDpco7rkJ2wqU5oL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "x=\\frac{3π}{4}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"$$\\csc\\left(x\\right)+\\sqrt{2}=0{\\quad:\\quad}x=\\frac{5π}{4}+2πn,\\:x=\\frac{7π}{4}+2πn$$", "input": "\\csc\\left(x\\right)+\\sqrt{2}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{2}\\:$$to the right side", "input": "\\csc\\left(x\\right)+\\sqrt{2}=0", "result": "\\csc\\left(x\\right)=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{2}$$ from both sides", "result": "\\csc\\left(x\\right)+\\sqrt{2}-\\sqrt{2}=0-\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "\\csc\\left(x\\right)=-\\sqrt{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\csc\\left(x\\right)=-\\sqrt{2}$$", "result": "x=\\frac{5π}{4}+2πn,\\:x=\\frac{7π}{4}+2πn", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\csc(x)&x&\\csc(x)\\\\\\hline 0&\\mathrm{Undefiend}&π&\\mathrm{Undefiend}\\\\\\hline \\frac{π}{6}&2&\\frac{7π}{6}&-2\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{4π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{5π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&\\sqrt{2}&\\frac{7π}{4}&-\\sqrt{2}\\\\\\hline \\frac{5π}{6}&2&\\frac{11π}{6}&-2\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{5π}{4}+2πn,\\:x=\\frac{7π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions csc 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\csc\\left(x\\right)-\\sqrt{2}=0{\\quad:\\quad}x=\\frac{π}{4}+2πn,\\:x=\\frac{3π}{4}+2πn$$", "input": "\\csc\\left(x\\right)-\\sqrt{2}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{2}\\:$$to the right side", "input": "\\csc\\left(x\\right)-\\sqrt{2}=0", "result": "\\csc\\left(x\\right)=\\sqrt{2}", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{2}$$ to both sides", "result": "\\csc\\left(x\\right)-\\sqrt{2}+\\sqrt{2}=0+\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "\\csc\\left(x\\right)=\\sqrt{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pF094G/W9zce+n3L6asoGi/eP8eIT34UG8omb/uZF6Y9i9oAvf9GdVXNbR3Y4FbqwGODcfcDyRNwANQkEpRh8P7OlGoLFf1X4hrgcqZHy85qe7fVTpcVcF3r+0EvCt4PK4jKt40vb8QaFNqOoFu5X60x+gh9gH6MihK9Pv1MXvnGaMZJe1ygjX89AS02/C9X3RaWPPkyUFEYdpRLdINt+PA3QMoN95NVoIQAARKufKBnsSNIkjoDn1lMCTXJvyjQuSue3TWVV3xkx9+AdeafQo7U2UN6des9vI5FFGUars8IdudjVm27ZGk/TTFPLJfKWMFM/Q22opjsN1GBGyQNsRlPWGFKFPK5+PyH2Eeh0ooXA8ImADsQWIDo7bl/ELcRaMlznQFCa8gKocMb0Z8pgOC1ZH/TxGrIUHTUaxRPfy1clIZfLzRJ3bRksUs1lMp48hXn+PJfFc3Uh1JOPr5wj0iZ3bB0BgUtwTNr3KnZpFWw14FmAV06QSwgd736/nVs5nTxhhX/JMpykzZI6J3VrKGoxMw626x3PqNkZlKxJbSBBTEk/JQ2cZ9WKuRzClU7XN+O5cJFrIQf6yTb4RLgAloHWB0IaORWdAGpGeqHb9WUbWSSTlA0mv7bnL8s2S/IvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "General solutions for $$\\csc\\left(x\\right)=\\sqrt{2}$$", "result": "x=\\frac{π}{4}+2πn,\\:x=\\frac{3π}{4}+2πn", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\csc(x)&x&\\csc(x)\\\\\\hline 0&\\mathrm{Undefiend}&π&\\mathrm{Undefiend}\\\\\\hline \\frac{π}{6}&2&\\frac{7π}{6}&-2\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{4π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{5π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&\\sqrt{2}&\\frac{7π}{4}&-\\sqrt{2}\\\\\\hline \\frac{5π}{6}&2&\\frac{11π}{6}&-2\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{4}+2πn,\\:x=\\frac{3π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions csc 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{4}+πn,\\:x=\\frac{3π}{4}+πn,\\:x=\\frac{5π}{4}+2πn,\\:x=\\frac{7π}{4}+2πn,\\:x=\\frac{π}{4}+2πn,\\:x=\\frac{3π}{4}+2πn" }, { "type": "step", "primary": "Merge Overlapping Intervals", "result": "x=\\frac{π}{4}+πn,\\: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)\\csc^{2}(x)-2\\cos(2x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }