prove (1+tan^2(x))/(2tan(x))=csc(2x)

{ "query": { "display": "prove $$\\frac{1+\\tan^{2}\\left(x\\right)}{2\\tan\\left(x\\right)}=\\csc\\left(2x\\right)$$", "symbolab_question": "TRIG_PROVING#prove \\frac{1+\\tan^{2}(x)}{2\\tan(x)}=\\csc(2x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Identities", "subTopic": "Other", "default": "\\mathrm{True}" }, "steps": { "type": "interim", "title": "Prove $$\\frac{1+\\tan^{2}\\left(x\\right)}{2\\tan\\left(x\\right)}=\\csc\\left(2x\\right):{\\quad}$$True", "input": "\\frac{1+\\tan^{2}\\left(x\\right)}{2\\tan\\left(x\\right)}=\\csc\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Manipulating left side", "secondary": [ "$$\\frac{1+\\tan^{2}\\left(x\\right)}{2\\tan\\left(x\\right)}$$" ] }, { "type": "interim", "title": "Express with sin, cos", "input": "\\frac{1+\\tan^{2}\\left(x\\right)}{2\\tan\\left(x\\right)}", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{2\\cos\\left(x\\right)\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=\\frac{1+\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}}{2\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}}" }, { "type": "interim", "title": "Simplify $$\\frac{1+\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}}{2\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}}:{\\quad}\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{2\\cos\\left(x\\right)\\sin\\left(x\\right)}$$", "input": "\\frac{1+\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}}{2\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}}", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{2\\cos\\left(x\\right)\\sin\\left(x\\right)}", "steps": [ { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\::{\\quad}\\frac{2\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "input": "2\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}", "result": "=\\frac{1+\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}}{\\frac{2\\sin\\left(x\\right)}{\\cos\\left(x\\right)}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sin\\left(x\\right)\\cdot\\:2}{\\cos\\left(x\\right)}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1+\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}}{\\frac{2\\sin\\left(x\\right)}{\\cos\\left(x\\right)}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{\\left(1+\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}\\right)\\cos\\left(x\\right)}{\\sin\\left(x\\right)\\cdot\\:2}" }, { "type": "interim", "title": "Join $$1+\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}:{\\quad}\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "1+\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}", "result": "=\\frac{\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}\\cos\\left(x\\right)}{2\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "result": "=\\frac{1\\cdot\\:\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}+\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\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\\:\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos^{2}\\left(x\\right)=\\cos^{2}\\left(x\\right)$$", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Multiply $$\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}\\cos\\left(x\\right)\\::{\\quad}\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}$$", "input": "\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}\\cos\\left(x\\right)", "result": "=\\frac{\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}}{2\\sin\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)\\right)\\cos\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\cos\\left(x\\right)$$", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)\\cdot\\:2}" } ], "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)}{2\\cos\\left(x\\right)\\sin\\left(x\\right)}", "result": "=\\frac{1}{\\sin\\left(2x\\right)}", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$2\\sin\\left(x\\right)\\cos\\left(x\\right)=\\sin\\left(2x\\right)$$", "result": "=\\frac{\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)}{\\sin\\left(2x\\right)}" }, { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "result": "=\\frac{1}{\\sin\\left(2x\\right)}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities", "result": "\\csc\\left(2x\\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(2x\\right)}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{\\frac{1}{\\csc\\left(2x\\right)}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{1}{\\frac{b}{c}}=\\frac{c}{b}$$", "result": "=\\frac{\\csc\\left(2x\\right)}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=\\csc\\left(2x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\csc\\left(2x\\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" } } }