(1-tan^2(A))/(1+tan^2(A))=1

{ "query": { "display": "$$\\frac{1-\\tan^{2}\\left(A\\right)}{1+\\tan^{2}\\left(A\\right)}=1$$", "symbolab_question": "EQUATION#\\frac{1-\\tan^{2}(A)}{1+\\tan^{2}(A)}=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "A=πn", "degrees": "A=0^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{1-\\tan^{2}\\left(A\\right)}{1+\\tan^{2}\\left(A\\right)}=1{\\quad:\\quad}A=πn$$", "input": "\\frac{1-\\tan^{2}\\left(A\\right)}{1+\\tan^{2}\\left(A\\right)}=1", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\frac{1-\\tan^{2}\\left(A\\right)}{1+\\tan^{2}\\left(A\\right)}=1", "result": "\\tan\\left(A\\right)=0", "steps": [ { "type": "step", "primary": "Let: $$\\tan\\left(A\\right)=u$$", "result": "\\frac{1-u^{2}}{1+u^{2}}=1" }, { "type": "interim", "title": "$$\\frac{1-u^{2}}{1+u^{2}}=1{\\quad:\\quad}u=0$$", "input": "\\frac{1-u^{2}}{1+u^{2}}=1", "steps": [ { "type": "interim", "title": "Multiply both sides by $$1+u^{2}$$", "input": "\\frac{1-u^{2}}{1+u^{2}}=1", "result": "1-u^{2}=1+u^{2}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$1+u^{2}$$", "result": "\\frac{1-u^{2}}{1+u^{2}}\\left(1+u^{2}\\right)=1\\cdot\\:\\left(1+u^{2}\\right)" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1-u^{2}}{1+u^{2}}\\left(1+u^{2}\\right)=1\\cdot\\:\\left(1+u^{2}\\right)", "result": "1-u^{2}=1+u^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1-u^{2}}{1+u^{2}}\\left(1+u^{2}\\right):{\\quad}1-u^{2}$$", "input": "\\frac{1-u^{2}}{1+u^{2}}\\left(1+u^{2}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(1-u^{2}\\right)\\left(1+u^{2}\\right)}{1+u^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$1+u^{2}$$", "result": "=1-u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vQiZXwqk1sTcgwvoCChnWdOeGW0gA+n4bDUyLqakaMOu7RYN6fcsKnw+vQ1xtlG1o5FYteSPKwXny4uCMrdsK2zhnS+ZG8fE60PSorn0+3dN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+SIbypcZIlHgt3hyCXgz8xzQvhG7T3SYTwryuCYHTVWw" } }, { "type": "interim", "title": "Simplify $$1\\cdot\\:\\left(1+u^{2}\\right):{\\quad}1+u^{2}$$", "input": "1\\cdot\\:\\left(1+u^{2}\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(1+u^{2}\\right)=\\left(1+u^{2}\\right)$$", "result": "=\\left(1+u^{2}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=1+u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mHkgMMCuSOAhxL6dLW5sna7tFg3p9ywqfD69DXG2UbWjkVi15I8rBefLi4Iyt2wrI3Tu9DCKN965iUkuba7BGU3kCh3oevUunZ7/b0qFKBQwRgti3uQ2HhPD7HmoaimpqAhmiTa5tCb9ot+tIfupNA==" } }, { "type": "step", "result": "1-u^{2}=1+u^{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve $$1-u^{2}=1+u^{2}:{\\quad}u=0$$", "input": "1-u^{2}=1+u^{2}", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1-u^{2}=1+u^{2}", "result": "-u^{2}=u^{2}", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1-u^{2}-1=1+u^{2}-1" }, { "type": "step", "primary": "Simplify", "result": "-u^{2}=u^{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$u^{2}\\:$$to the left side", "input": "-u^{2}=u^{2}", "result": "-2u^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$u^{2}$$ from both sides", "result": "-u^{2}-u^{2}=u^{2}-u^{2}" }, { "type": "step", "primary": "Simplify", "result": "-2u^{2}=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2u^{2}=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2u^{2}=0", "result": "u^{2}=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2u^{2}}{-2}=\\frac{0}{-2}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq" } }, { "type": "step", "result": "u=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(A\\right)$$", "result": "\\tan\\left(A\\right)=0" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\tan\\left(A\\right)=0{\\quad:\\quad}A=πn$$", "input": "\\tan\\left(A\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(A\\right)=0$$", "result": "A=0+πn", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "A=0+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$A=0+πn:{\\quad}A=πn$$", "input": "A=0+πn", "steps": [ { "type": "step", "primary": "$$0+πn=πn$$", "result": "A=πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "A=πn" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "A=πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "A", "plotRequest": "\\frac{1-\\tan^{2}(A)}{1+\\tan^{2}(A)}-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }