prove cos(x)cot(x)+sin(x)=csc(x)

{ "query": { "display": "prove $$\\cos\\left(x\\right)\\cot\\left(x\\right)+\\sin\\left(x\\right)=\\csc\\left(x\\right)$$", "symbolab_question": "TRIG_PROVING#prove \\cos(x)\\cot(x)+\\sin(x)=\\csc(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Identities", "subTopic": "Other", "default": "\\mathrm{True}" }, "steps": { "type": "interim", "title": "Prove $$\\cos\\left(x\\right)\\cot\\left(x\\right)+\\sin\\left(x\\right)=\\csc\\left(x\\right):{\\quad}$$True", "input": "\\cos\\left(x\\right)\\cot\\left(x\\right)+\\sin\\left(x\\right)=\\csc\\left(x\\right)", "steps": [ { "type": "step", "primary": "Manipulating left side", "secondary": [ "$$\\cos\\left(x\\right)\\cot\\left(x\\right)+\\sin\\left(x\\right)$$" ] }, { "type": "interim", "title": "Express with sin, cos", "input": "\\sin\\left(x\\right)+\\cos\\left(x\\right)\\cot\\left(x\\right)", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\cot\\left(x\\right)=\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=\\sin\\left(x\\right)+\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "Simplify $$\\sin\\left(x\\right)+\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}:{\\quad}\\frac{\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\sin\\left(x\\right)+\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "result": "=\\frac{\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}=\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}$$", "input": "\\cos\\left(x\\right)\\frac{\\cos\\left(x\\right)}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\cos\\left(x\\right)\\cos\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{1+1}\\left(x\\right)$$" ], "result": "=\\cos^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\cos^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw47oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJVxXBxD1gYRAlNp97nQuTZFXRu5R8U1G8Rh9s+llHwfqtic1bCnH3jLV3vr22vWk8gIJE6eFSdaQPkT4FMktmcw==" } }, { "type": "step", "result": "=\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BomPtzG+UOWoCHKw/+Shmm5AZDqQW6CzLlif7pmGqWo8h+u5fV4iBAkECPyXf7J5zMFYmi1F5Hg/ibpEToVnYwDmEzFfqMSLndLqC5qAp4dwPIzt1oWxMWVTfovlFJboZEt3ZXAiqUE0HIXrrrezJH7g6cH2OfSl85iIuybAtg4M9I0lFOxOZYaH32mNX0s3xGpk+gGxnOLtRja0BBQnbsUslKYonmvF+wmNIbKyxLWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\sin\\left(x\\right)+\\frac{\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Convert element to fraction: $$\\sin\\left(x\\right)=\\frac{\\sin\\left(x\\right)\\sin\\left(x\\right)}{\\sin\\left(x\\right)}$$", "result": "=\\frac{\\sin\\left(x\\right)\\sin\\left(x\\right)}{\\sin\\left(x\\right)}+\\frac{\\cos^{2}\\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{\\sin\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)=\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)$$", "input": "\\sin\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(x\\right)\\sin\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "input": "\\sin\\left(x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sin\\left(x\\right)\\sin\\left(x\\right)=\\:\\sin^{1+1}\\left(x\\right)$$" ], "result": "=\\sin^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\sin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Db5kOPGdwaYYCr65H3kEuI7oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJh8ihxP+4PwTZMAHoWYe0rYRgj2SDWqhbeg4ibDhNi7JNro/AJAWcGEjut/HzR49zgXIiLmXebpqW8NAeupm/ZQ==" } }, { "type": "step", "result": "=\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Db5kOPGdwaYYCr65H3kEuLiZqVAVQQuMwm9zVxbuK7UtOtZYwUjyXhDTsNnn6ElrPg47u1resbhMeEwTUR/+zmDuL/+6ie9rzeVbiKZ0CzRRfI1Hc6iuEfTvbINdgvBoJ/oRjOB+b1a4vJ5oIBXyWZp0GdLahlxcloLqWlCBNWR6z4uIaWtVXUsQ3uow1BnXaO0fmhEbNRGeME2Cf7U6KQ==" } }, { "type": "step", "result": "=\\frac{\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)}{\\sin\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Express Sin Cos 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\sin\\left(x\\right)}", "result": "=\\frac{1}{\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "result": "=\\frac{1}{\\sin\\left(x\\right)}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities", "result": "\\csc\\left(x\\right)", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\sin\\left(x\\right)=\\frac{1}{\\csc\\left(x\\right)}$$", "result": "\\frac{1}{\\frac{1}{\\csc\\left(x\\right)}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{\\frac{1}{\\csc\\left(x\\right)}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{1}{\\frac{b}{c}}=\\frac{c}{b}$$", "result": "=\\frac{\\csc\\left(x\\right)}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=\\csc\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\csc\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "primary": "We showed that the two sides could take the same form", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "solvingClass": "Trig Proving", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Identities", "practiceTopic": "Trig Identities" } } }