csc(x)= 1/(sec(x))

{ "query": { "display": "$$\\csc\\left(x\\right)=\\frac{1}{\\sec\\left(x\\right)}$$", "symbolab_question": "EQUATION#\\csc(x)=\\frac{1}{\\sec(x)}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\csc\\left(x\\right)=\\frac{1}{\\sec\\left(x\\right)}{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\csc\\left(x\\right)=\\frac{1}{\\sec\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{1}{\\sec\\left(x\\right)}$$ from both sides", "result": "\\csc\\left(x\\right)-\\frac{1}{\\sec\\left(x\\right)}=0" }, { "type": "interim", "title": "Simplify $$\\csc\\left(x\\right)-\\frac{1}{\\sec\\left(x\\right)}:{\\quad}\\frac{\\csc\\left(x\\right)\\sec\\left(x\\right)-1}{\\sec\\left(x\\right)}$$", "input": "\\csc\\left(x\\right)-\\frac{1}{\\sec\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\csc\\left(x\\right)=\\frac{\\csc\\left(x\\right)\\sec\\left(x\\right)}{\\sec\\left(x\\right)}$$", "result": "=\\frac{\\csc\\left(x\\right)\\sec\\left(x\\right)}{\\sec\\left(x\\right)}-\\frac{1}{\\sec\\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{\\csc\\left(x\\right)\\sec\\left(x\\right)-1}{\\sec\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{\\csc\\left(x\\right)\\sec\\left(x\\right)-1}{\\sec\\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": "\\csc\\left(x\\right)\\sec\\left(x\\right)-1=0" }, { "type": "interim", "title": "Express with sin, cos", "input": "-1+\\csc\\left(x\\right)\\sec\\left(x\\right)", "result": "\\frac{1-\\cos\\left(x\\right)\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)}=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\csc\\left(x\\right)=\\frac{1}{\\sin\\left(x\\right)}$$", "result": "=-1+\\frac{1}{\\sin\\left(x\\right)}\\sec\\left(x\\right)" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\sec\\left(x\\right)=\\frac{1}{\\cos\\left(x\\right)}$$", "result": "=-1+\\frac{1}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$-1+\\frac{1}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}:{\\quad}\\frac{-\\sin\\left(x\\right)\\cos\\left(x\\right)+1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}$$", "input": "-1+\\frac{1}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}", "result": "=\\frac{-\\sin\\left(x\\right)\\cos\\left(x\\right)+1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}=\\frac{1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}$$", "input": "\\frac{1}{\\sin\\left(x\\right)}\\cdot\\:\\frac{1}{\\cos\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s796u7EfZY/A0BtgZqQ7GW1/jrRH4moVOMvUWsavbquQ8sQSPOA76boc8p9C9laP9ELTrWWMFI8l4Q07DZ5+hJa0hqcfXty6JYd1pWIgQ3ZjhP40FLK0hyDQk6txhGNSz+kEoMOb36Fi0gAPlSQGZoDv/829doud4baah5J7nhH2gotuW0XAImOrM+0dw8ryGakQMKgOzcRr+YyVH76qFrNFbKBIKuoMBuTl/c/oup5ItmRco9E7Z+mCbF1Xfz6gFRsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=-1+\\frac{1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\sin\\left(x\\right)\\cos\\left(x\\right)}{\\sin\\left(x\\right)\\cos\\left(x\\right)}$$", "result": "=-\\frac{1\\cdot\\:\\sin\\left(x\\right)\\cos\\left(x\\right)}{\\sin\\left(x\\right)\\cos\\left(x\\right)}+\\frac{1}{\\sin\\left(x\\right)\\cos\\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{-1\\cdot\\:\\sin\\left(x\\right)\\cos\\left(x\\right)+1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sin\\left(x\\right)=\\sin\\left(x\\right)$$", "result": "=\\frac{-\\sin\\left(x\\right)\\cos\\left(x\\right)+1}{\\sin\\left(x\\right)\\cos\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Express Sin Cos 0Eq" } }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "1-\\cos\\left(x\\right)\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "1-\\cos\\left(x\\right)\\sin\\left(x\\right)", "result": "1-\\frac{\\sin\\left(2x\\right)}{2}=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$2\\sin\\left(x\\right)\\cos\\left(x\\right)=\\sin\\left(2x\\right)$$", "secondary": [ "$$\\sin\\left(x\\right)\\cos\\left(x\\right)=\\frac{\\sin\\left(2x\\right)}{2}$$" ], "result": "=1-\\frac{\\sin\\left(2x\\right)}{2}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PUOM60ShVE7f9cgKGM2tAUJfFQ0M/5QJI9z++Ql2FlGyuRcnFgnh8AijhYNjhqyRbyZYyX6EGMAcAMgjOADqPLajIxP2OrjkBCECQd8a6w6Sr+mcmYnVxTKJ/0EhrUie5GkthY25pxn0BviQhgwp53Sb8BBwyqm1RVzhx3vZiieIP2TH9VLmoizBQwdUXcYEMwlmQv/qPmO1VQ4glIE4qqXFidbElO7XH8+PPiEMftA=" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1-\\frac{\\sin\\left(2x\\right)}{2}=0", "result": "-\\frac{\\sin\\left(2x\\right)}{2}=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1-\\frac{\\sin\\left(2x\\right)}{2}-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{\\sin\\left(2x\\right)}{2}=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "-\\frac{\\sin\\left(2x\\right)}{2}=-1", "result": "-\\sin\\left(2x\\right)=-2", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "2\\left(-\\frac{\\sin\\left(2x\\right)}{2}\\right)=2\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\sin\\left(2x\\right)=-2" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-\\sin\\left(2x\\right)=-2", "result": "\\sin\\left(2x\\right)=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-\\sin\\left(2x\\right)}{-1}=\\frac{-2}{-1}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(2x\\right)=2" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\csc(x)-\\frac{1}{\\sec(x)}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }