1/(1-cos(x))+1/(1+cos(x))=2csc(x)

{ "query": { "display": "$$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right)$$", "symbolab_question": "EQUATION#\\frac{1}{1-\\cos(x)}+\\frac{1}{1+\\cos(x)}=2\\csc(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right){\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$2\\csc\\left(x\\right)$$ from both sides", "result": "\\frac{2\\cos^{2}\\left(x\\right)\\csc\\left(x\\right)-2\\csc\\left(x\\right)+2}{\\left(-\\cos\\left(x\\right)+1\\right)\\left(\\cos\\left(x\\right)+1\\right)}=0" }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "2\\cos^{2}\\left(x\\right)\\csc\\left(x\\right)-2\\csc\\left(x\\right)+2=0" }, { "type": "step", "primary": "Express with sin, cos", "result": "2\\cos^{2}\\left(x\\right)\\frac{1}{\\sin\\left(x\\right)}-2\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}+2=0" }, { "type": "interim", "title": "Simplify $$2\\cos^{2}\\left(x\\right)\\frac{1}{\\sin\\left(x\\right)}-2\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}+2:{\\quad}\\frac{2\\cos^{2}\\left(x\\right)-2+2\\sin\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "2\\cos^{2}\\left(x\\right)\\frac{1}{\\sin\\left(x\\right)}-2\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}+2", "steps": [ { "type": "interim", "title": "$$2\\cos^{2}\\left(x\\right)\\frac{1}{\\sin\\left(x\\right)}=\\frac{2\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "2\\cos^{2}\\left(x\\right)\\frac{1}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FMDrC3J/ZVrenAVm+Grw5qVKrROcmlcTz7XT3cJVLXGDF0/IXXx1QUFyFLR7eQUV/aL2Coo0GISQwm8bv5wDiYaC0pJBc5B8ILrb6YkqrjlniYK0O2N2Mz98K2t68ccgPPnqwLUhBdmpRB9XjrjnBg+t++sqZb7nkUyR5Ny4dRzXtqvY3UqkOfNNkTOzftT+hoLSkkFzkHwgutvpiSquOb5GFzK2LTZa81zLGLoN2V4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}=\\frac{2}{\\sin\\left(x\\right)}$$", "input": "2\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviuH/Jf40TyR+hfPSSAWLHqrehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpghGOfwTqN/m5GliOrDUVmJ04xTNlEc8iWnVHxPamSnbS8PgbRY6lX6tcMqfLeRhCy2GVTF4gOJZqTVXdY8hNmv/XzFYr0gCk5ceNVISXD1O" } }, { "type": "step", "result": "=\\frac{2\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}-\\frac{2}{\\sin\\left(x\\right)}+2" }, { "type": "interim", "title": "Combine the fractions $$\\frac{2\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}-\\frac{2}{\\sin\\left(x\\right)}:{\\quad}\\frac{2\\cos^{2}\\left(x\\right)-2}{\\sin\\left(x\\right)}$$", "result": "=\\frac{2\\cos^{2}\\left(x\\right)-2}{\\sin\\left(x\\right)}+2", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2\\cos^{2}\\left(x\\right)-2}{\\sin\\left(x\\right)}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2\\sin\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=\\frac{2\\cos^{2}\\left(x\\right)-2}{\\sin\\left(x\\right)}+\\frac{2\\sin\\left(x\\right)}{\\sin\\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{2\\cos^{2}\\left(x\\right)-2+2\\sin\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{2\\cos^{2}\\left(x\\right)-2+2\\sin\\left(x\\right)}{\\sin\\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": "2\\cos^{2}\\left(x\\right)-2+2\\sin\\left(x\\right)=0" }, { "type": "step", "primary": "Subtract $$2\\sin\\left(x\\right)$$ from both sides", "result": "2\\cos^{2}\\left(x\\right)-2=-2\\sin\\left(x\\right)" }, { "type": "step", "primary": "Square both sides", "result": "\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}=\\left(-2\\sin\\left(x\\right)\\right)^{2}" }, { "type": "step", "primary": "Subtract $$\\left(-2\\sin\\left(x\\right)\\right)^{2}$$ from both sides", "result": "\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-4\\sin^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Factor $$\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-4\\sin^{2}\\left(x\\right):{\\quad}4\\left(\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)\\right)\\left(\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)\\right)$$", "input": "\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-4\\sin^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "Rewrite $$\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-4\\sin^{2}\\left(x\\right)$$ as $$\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-\\left(2\\sin\\left(x\\right)\\right)^{2}$$", "input": "\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-4\\sin^{2}\\left(x\\right)", "result": "=\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-\\left(2\\sin\\left(x\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Rewrite $$4$$ as $$2^{2}$$", "result": "=\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-2^{2}\\sin^{2}\\left(x\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$2^{2}\\sin^{2}\\left(x\\right)=\\left(2\\sin\\left(x\\right)\\right)^{2}$$" ], "result": "=\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-\\left(2\\sin\\left(x\\right)\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(2\\cos^{2}\\left(x\\right)-2\\right)^{2}-\\left(2\\sin\\left(x\\right)\\right)^{2}=\\left(\\left(2\\cos^{2}\\left(x\\right)-2\\right)+2\\sin\\left(x\\right)\\right)\\left(\\left(2\\cos^{2}\\left(x\\right)-2\\right)-2\\sin\\left(x\\right)\\right)$$" ], "result": "=\\left(\\left(2\\cos^{2}\\left(x\\right)-2\\right)+2\\sin\\left(x\\right)\\right)\\left(\\left(2\\cos^{2}\\left(x\\right)-2\\right)-2\\sin\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } }, { "type": "step", "primary": "Refine", "result": "=\\left(2\\cos^{2}\\left(x\\right)+2\\sin\\left(x\\right)-2\\right)\\left(2\\cos^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2\\right)" }, { "type": "interim", "title": "Factor $$2\\cos^{2}\\left(x\\right)-2+2\\sin\\left(x\\right):{\\quad}2\\left(\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)\\right)$$", "input": "2\\cos^{2}\\left(x\\right)-2+2\\sin\\left(x\\right)", "result": "=2\\left(\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)-1\\right)\\left(2\\cos^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$2\\cos^{2}\\left(x\\right)-2-2\\sin\\left(x\\right):{\\quad}2\\left(\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)\\right)$$", "input": "2\\cos^{2}\\left(x\\right)-2-2\\sin\\left(x\\right)", "result": "=2\\left(\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)\\right)\\cdot\\:2\\left(\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=4\\left(\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)\\right)\\left(\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "4\\left(\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)\\right)\\left(\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)=0\\lor\\:\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "$$\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+2πn$$", "input": "\\cos^{2}\\left(x\\right)-1+\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos^{2}\\left(x\\right)+\\sin\\left(x\\right)", "result": "\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1=\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "secondary": [ "$$1-\\cos^{2}\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "$$-1+\\cos^{2}\\left(x\\right)=-\\sin^{2}\\left(x\\right)$$" ], "result": "=\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VCH+prMD3oJfvXUn3MkBUiKk+FPosJui0EvbDbx83/CqPO0XSKqaWlOK6mHDvVM10YIYWgyd6SbL0eRRsTHoy/YnSKF5/4+51qVY0U4KnLmxpmQ3E46bONVTEqOrZvquzu2Oe4S4Uqbap8RX3jWL1jG9dDZkXCs5+Cxg1YNvJNPUe9yYF2Q5qSv3gDlLQoBFjj5yGUAwgawpYTH/Vvtg3vE=" } }, { "type": "interim", "title": "Solve by substitution", "input": "\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u-u^{2}=0" }, { "type": "interim", "title": "$$u-u^{2}=0{\\quad:\\quad}u=0,\\:u=1$$", "input": "u-u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-u^{2}+u=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-u^{2}+u=0", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:0}}{2\\left(-1\\right)}", "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 $${\\quad}a=-1,\\:b=1,\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:0}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QNeNf/Evw3xC3YbFt1R0HoieIz0UW+uwk8JH6MfJEEjtdd2YCGrLNonW+vEDv/fSanoZPShdwJWDv/wN7YGbQPVP9JoHOzFn9f3kpOEpmga8beBAdyJ9jsFRvCHiN82OQ4Sl9HRs579T5/4k/bbKVqyAqtQY4SvE0Qb4wO5tywXHXumI6JsRbJqSendMBbfjiS267MvT1JLdT5GyDEcYVcz8sCqFW4KRMi5zptad+UisoAiCZLxb22KZVtGIU1iNDGcUXRLJTTa+dDKvHVOi77R2oHZlFG8d7jRjCNpMxGMtt47Cfbjwk5Ca+gEVzKSfj9HwPEKjie0I7TPaY/pZ1IxpS5BM0jcZOoblVmqSrgjeuhyPJkr6H5MBaATAuti4crnz9yuc4EsY0b0w2yPlm1ZduS7VBg2CeuvAXrT5UyEYeD3v3tLMm4gnJHIJ0xqxWf5QhIGPdofhXkud0XCXJTA71ZZ9CQJrKI4aKG+rI5pioBB07qOCG/zzk7H6foTCDJf8+s9Gl86Ap3MdcDJI0Jtnyq/Im3tuQKRMtvslXszggwh9g3AzYmpTLr8uwZ8J5PlGPSQROb/2EmerPhnM852KpwArzUqjj8qnIU+IJnx2m2cIuScrTf9bn5kIkoppBjIEkvaPTvlHoUdWt09SE1p2DDO5jolvRyiBIh4pWq5RxuKcrXgaifl/dAFyf8kqF0Fpmjpw0ipi1HAoepUIU0hov4V+wJdL8XMVz+rDJZASQ8kl9I0ZXtQcoEX4aTKe" } }, { "type": "interim", "title": "$$\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:0}=1$$", "input": "\\sqrt{1^{2}-4\\left(-1\\right)\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\left(-1\\right)\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{1+4\\cdot\\:1\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{1+0}" }, { "type": "step", "primary": "Add the numbers: $$1+0=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uac2nmABaurcWf8xH8OQy0STQbxvEVHwQywLP/SJJgPehkKrn0era9rz8TlL+x/vj+lG4r/2yK6aJdxvUCCu4btCR5dIjxQ5ASg+ZPFVSseQYNPYZR4RrqS3lX81H+QPZyb36vpKwrPyssOZsvwtrg==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-1+1}{2\\left(-1\\right)},\\:{u}_{2}=\\frac{-1-1}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-1+1}{2\\left(-1\\right)}:{\\quad}0$$", "input": "\\frac{-1+1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1+1}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+1=0$$", "result": "=\\frac{0}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{0}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZILxh0n9XGu+UX30rV52Ka15SZXhTlhKvYvwHG+unGRwkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX6QS8+Ejzws6A1XwOMup5uKD9q1+pR1ccvpxEymyUlf8" } }, { "type": "interim", "title": "$$u=\\frac{-1-1}{2\\left(-1\\right)}:{\\quad}1$$", "input": "\\frac{-1-1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1-1}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=\\frac{-2}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-2}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CoFrCplKlG9JQtC7YmAoI615SZXhTlhKvYvwHG+unGRwkKGJWEPFPk38sdJMsyPIeqXfySbC6vm4UawE43QWXVMyKerRJX9uXZpr8ibTE6CD9q1+pR1ccvpxEymyUlf8" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=0,\\:u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "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": "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": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)=0{\\quad:\\quad}x=\\frac{3π}{2}+2πn,\\:x=2πn,\\:x=π+2πn$$", "input": "\\cos^{2}\\left(x\\right)-1-\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-1+\\cos^{2}\\left(x\\right)-\\sin\\left(x\\right)", "result": "-\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1=\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "secondary": [ "$$1-\\cos^{2}\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "$$-1+\\cos^{2}\\left(x\\right)=-\\sin^{2}\\left(x\\right)$$" ], "result": "=-\\sin\\left(x\\right)-\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XQoqDkagBdkaGF3BP1M6VHH7uP/mVFxkT/o/Bd7w+dSk+FPosJui0EvbDbx83/CqPO0XSKqaWlOK6mHDvVM10YIYWgyd6SbL0eRRsTHoy/YnSKF5/4+51qVY0U4KnLmxyT0+JDteYKi5dDP8qH9sz0ToXaufgATy26ksCzoJvs5N5Aod6Hr1Lp2e/29KhSgUwjzZVGI+drXOGPAfNCN9AeZetBcqyKSqUvKkqZYGMTwkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Solve by substitution", "input": "-\\sin\\left(x\\right)-\\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": "-u-u^{2}=0" }, { "type": "interim", "title": "$$-u-u^{2}=0{\\quad:\\quad}u=-1,\\:u=0$$", "input": "-u-u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-u^{2}-u=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-u^{2}-u=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}}{2\\left(-1\\right)}", "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 $${\\quad}a=-1,\\:b=-1,\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}=1$$", "input": "\\sqrt{\\left(-1\\right)^{2}-4\\left(-1\\right)\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-1\\right)^{2}+4\\cdot\\:1\\cdot\\:0}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:0=0$$", "input": "4\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7SD/swv2EbFFv+X+7iEc3n+jkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhFxXGJQXf9dTesK2NoC90LlXeV03GYdXLZ3ZnsTzJuK6" } }, { "type": "step", "result": "=\\sqrt{1+0}" }, { "type": "step", "primary": "Add the numbers: $$1+0=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gcCSkwHGkSzBJS28+axO6/ozfpat709n1xtF07NKaGIAlilG71elit3w1IBbYN0PfzBJSNoxCO6fFy6iRIjrm6N6Hv6MoTMtvtU0IQwXdn+SVpPUu8d2DohT7uf7kqbJfQtm3YLrBni/HT2B8g+NXiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-1\\right)+1}{2\\left(-1\\right)},\\:{u}_{2}=\\frac{-\\left(-1\\right)-1}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-1\\right)+1}{2\\left(-1\\right)}:{\\quad}-1$$", "input": "\\frac{-\\left(-1\\right)+1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1+1}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{2}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73nEuax+m66TuGCt9MmWfwONiDKCNsnmWP0fTjbhukYR1g99dC9fj9sg0EHzBIRDRlcq1iPPbKQKUi0Yqft4tTnXxzR/D3xpyR5yXTZ2YQF0oWmfDawUG2OTsqk0PdSrxvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-1\\right)-1}{2\\left(-1\\right)}:{\\quad}0$$", "input": "\\frac{-\\left(-1\\right)-1}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1-1}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$1-1=0$$", "result": "=\\frac{0}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{0}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71Z4tBeCEgm6o48U48XbdAONiDKCNsnmWP0fTjbhukYR1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl8XKhRRe6+fuRKwL9f/rSxRwSdSDkhMgABQT7Jz4EHtrJLd1ohke2Wgml78++2zI0g==" } }, { "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": "$$\\sin\\left(x\\right)=-1{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{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{3π}{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{3π}{2}+2πn,\\:x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" }, { "type": "interim", "title": "Verify solutions by plugging them into the original equation", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right)$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$2πn:{\\quad}$$False", "input": "2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "2π1" }, { "type": "step", "primary": "For $$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right){\\quad}$$plug in$${\\quad}x=2π1$$", "result": "\\frac{1}{1-\\cos\\left(2π1\\right)}+\\frac{1}{1+\\cos\\left(2π1\\right)}=2\\csc\\left(2π1\\right)" }, { "type": "step", "result": "\\mathrm{Undefined}" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$π+2πn:{\\quad}$$False", "input": "π+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "π+2π1" }, { "type": "step", "primary": "For $$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right){\\quad}$$plug in$${\\quad}x=π+2π1$$", "result": "\\frac{1}{1-\\cos\\left(π+2π1\\right)}+\\frac{1}{1+\\cos\\left(π+2π1\\right)}=2\\csc\\left(π+2π1\\right)" }, { "type": "step", "result": "\\mathrm{Undefined}" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$\\frac{π}{2}+2πn:{\\quad}$$True", "input": "\\frac{π}{2}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{π}{2}+2π1" }, { "type": "step", "primary": "For $$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right){\\quad}$$plug in$${\\quad}x=\\frac{π}{2}+2π1$$", "result": "\\frac{1}{1-\\cos\\left(\\frac{π}{2}+2π1\\right)}+\\frac{1}{1+\\cos\\left(\\frac{π}{2}+2π1\\right)}=2\\csc\\left(\\frac{π}{2}+2π1\\right)" }, { "type": "step", "primary": "Refine", "result": "2=2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$\\frac{3π}{2}+2πn:{\\quad}$$False", "input": "\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{3π}{2}+2π1" }, { "type": "step", "primary": "For $$\\frac{1}{1-\\cos\\left(x\\right)}+\\frac{1}{1+\\cos\\left(x\\right)}=2\\csc\\left(x\\right){\\quad}$$plug in$${\\quad}x=\\frac{3π}{2}+2π1$$", "result": "\\frac{1}{1-\\cos\\left(\\frac{3π}{2}+2π1\\right)}+\\frac{1}{1+\\cos\\left(\\frac{3π}{2}+2π1\\right)}=2\\csc\\left(\\frac{3π}{2}+2π1\\right)" }, { "type": "step", "primary": "Refine", "result": "2=-2" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "result": "x=\\frac{π}{2}+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{1}{1-\\cos(x)}+\\frac{1}{1+\\cos(x)}-2\\csc(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }