(tan(60)-tan(45))/(1+tan(60)tan(45))

{ "query": { "display": "$$\\frac{\\tan\\left(60^{\\circ\\:}\\right)-\\tan\\left(45^{\\circ\\:}\\right)}{1+\\tan\\left(60^{\\circ\\:}\\right)\\tan\\left(45^{\\circ\\:}\\right)}$$", "symbolab_question": "TRIG_EVALUATE#\\frac{\\tan(60^{\\circ })-\\tan(45^{\\circ })}{1+\\tan(60^{\\circ })\\tan(45^{\\circ })}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "2-\\sqrt{3}", "decimal": "0.26794…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\tan\\left(60^{\\circ\\:}\\right)-\\tan\\left(45^{\\circ\\:}\\right)}{1+\\tan\\left(60^{\\circ\\:}\\right)\\tan\\left(45^{\\circ\\:}\\right)}=2-\\sqrt{3}$$", "input": "\\frac{\\tan\\left(60^{\\circ\\:}\\right)-\\tan\\left(45^{\\circ\\:}\\right)}{1+\\tan\\left(60^{\\circ\\:}\\right)\\tan\\left(45^{\\circ\\:}\\right)}", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\tan\\left(60^{\\circ\\:}\\right)=\\sqrt{3}$$", "input": "\\tan\\left(60^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\sqrt{3}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\tan\\left(45^{\\circ\\:}\\right)=1$$", "input": "\\tan\\left(45^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}-1}{1+\\sqrt{3}\\cdot\\:1}" }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{3}-1}{1+\\sqrt{3}\\cdot\\:1}:{\\quad}2-\\sqrt{3}$$", "input": "\\frac{\\sqrt{3}-1}{1+\\sqrt{3}\\cdot\\:1}", "result": "=2-\\sqrt{3}", "steps": [ { "type": "step", "primary": "Multiply: $$\\sqrt{3}\\cdot\\:1=\\sqrt{3}$$", "result": "=\\frac{\\sqrt{3}-1}{1+\\sqrt{3}}" }, { "type": "interim", "title": "Rationalize $$\\frac{\\sqrt{3}-1}{1+\\sqrt{3}}:{\\quad}2-\\sqrt{3}$$", "input": "\\frac{\\sqrt{3}-1}{1+\\sqrt{3}}", "result": "=2-\\sqrt{3}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{1-\\sqrt{3}}{1-\\sqrt{3}}$$", "result": "=\\frac{\\left(\\sqrt{3}-1\\right)\\left(1-\\sqrt{3}\\right)}{\\left(1+\\sqrt{3}\\right)\\left(1-\\sqrt{3}\\right)}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$1+\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}-1\\right)\\left(1-\\sqrt{3}\\right)=2\\sqrt{3}-4$$", "input": "\\left(\\sqrt{3}-1\\right)\\left(1-\\sqrt{3}\\right)", "steps": [ { "type": "step", "primary": "Apply FOIL method: $$\\left(a+b\\right)\\left(c+d\\right)=ac+ad+bc+bd$$", "secondary": [ "$$a=\\sqrt{3},\\:b=-1,\\:c=1,\\:d=-\\sqrt{3}$$" ], "result": "=\\sqrt{3}\\cdot\\:1+\\sqrt{3}\\left(-\\sqrt{3}\\right)+\\left(-1\\right)\\cdot\\:1+\\left(-1\\right)\\left(-\\sqrt{3}\\right)", "meta": { "title": { "extension": "F-First<br/>O-Outer<br/>I-Inner<br/>L-Last" }, "practiceLink": "/practice/expansion-practice#area=main&subtopic=FOIL_Basic", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a,\\:\\:\\left(-a\\right)\\left(-b\\right)=ab$$" ], "result": "=1\\cdot\\:\\sqrt{3}-\\sqrt{3}\\sqrt{3}-1\\cdot\\:1+1\\cdot\\:\\sqrt{3}" }, { "type": "interim", "title": "Simplify $$1\\cdot\\:\\sqrt{3}-\\sqrt{3}\\sqrt{3}-1\\cdot\\:1+1\\cdot\\:\\sqrt{3}:{\\quad}2\\sqrt{3}-4$$", "input": "1\\cdot\\:\\sqrt{3}-\\sqrt{3}\\sqrt{3}-1\\cdot\\:1+1\\cdot\\:\\sqrt{3}", "result": "=2\\sqrt{3}-4", "steps": [ { "type": "step", "primary": "Add similar elements: $$1\\cdot\\:\\sqrt{3}+1\\cdot\\:\\sqrt{3}=2\\sqrt{3}$$", "result": "=2\\sqrt{3}-\\sqrt{3}\\sqrt{3}-1\\cdot\\:1" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=2\\sqrt{3}-3-1\\cdot\\:1", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=2\\sqrt{3}-3-1" }, { "type": "step", "primary": "Subtract the numbers: $$-3-1=-4$$", "result": "=2\\sqrt{3}-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72Tj1eRE79IZYtdTGOh0xx9WJQ8KUvYiN3PjaRYPbA1EgJ/ZZA32ZInFBpDtxBfiKBk/ikpVmbgslT1cIfmn62Ij14fl7W4WE/18O5OMWC5nMpfN7WBEZJCvLPqWdHo6cKfjf7uTkTc65s4FUZxy+cZf2c8Pmaw7a3P0kFkh40Xk=" } }, { "type": "interim", "title": "$$\\left(1+\\sqrt{3}\\right)\\left(1-\\sqrt{3}\\right)=-2$$", "input": "\\left(1+\\sqrt{3}\\right)\\left(1-\\sqrt{3}\\right)", "result": "=\\frac{2\\sqrt{3}-4}{-2}", "steps": [ { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$\\left(a+b\\right)\\left(a-b\\right)=a^{2}-b^{2}$$", "secondary": [ "$$a=1,\\:b=\\sqrt{3}$$" ], "result": "=1^{2}-\\left(\\sqrt{3}\\right)^{2}", "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Expand Difference of Squares" } }, { "type": "interim", "title": "Simplify $$1^{2}-\\left(\\sqrt{3}\\right)^{2}:{\\quad}-2$$", "input": "1^{2}-\\left(\\sqrt{3}\\right)^{2}", "result": "=-2", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=1-\\left(\\sqrt{3}\\right)^{2}" }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}=3$$", "input": "\\left(\\sqrt{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XEzlndel0x/QlMW8VTD5UXyRHuGw7+tM5METTDj6vVEMSTe+hEEEr2+K9b3W9JkG8SrqrDW4mFcEK+hPNqZN8jCFrm8vCvLZxZsdY4NLQVuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=1-3" }, { "type": "step", "primary": "Subtract the numbers: $$1-3=-2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wGST/+4mwV+axGbbDZTT1dWJQ8KUvYiN3PjaRYPbA1EgJ/ZZA32ZInFBpDtxBfiK7/W/ggfXH4imxtjzN0Y14P6yde82fOG9s1dRW1cJPoC0PmpfYrwHyTPp6/mKbzqhnm3YHWp9NFLxPpoC/WvthQ==" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "secondary": [ "$$2\\sqrt{3}-4=-\\left(4-2\\sqrt{3}\\right)$$" ], "result": "=\\frac{4-2\\sqrt{3}}{2}" }, { "type": "interim", "title": "Factor $$4-2\\sqrt{3}:{\\quad}2\\left(2-\\sqrt{3}\\right)$$", "input": "4-2\\sqrt{3}", "result": "=\\frac{2\\left(2-\\sqrt{3}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:2-2\\sqrt{3}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(2-\\sqrt{3}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=2-\\sqrt{3}" } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMVduaPyBreClJYYjEvD6Y6li3zFt1xK+8FxKJk2Zgoo5zRqDxPUzBN6vjj5oJL9kUOdNXLAv9KuJwiUjYLx+LesUmHg+iQxGlGxKWDb8ymJzHimBRYRqHSWeJkuUPhfTC/j+KYYgjJt/QT9RFe9c9dNcdqYhHvqFxntw3eancoJbNGT+IKBu4xZ/O6OQHcwSGA==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }