(sec(pi/4))^2

{ "query": { "display": "$$\\left(\\sec\\left(\\frac{π}{4}\\right)\\right)^{2}$$", "symbolab_question": "TRIG_EVALUATE#(\\sec(\\frac{π}{4}))^{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(\\sec\\left(\\frac{π}{4}\\right)\\right)^{2}=2$$", "input": "\\left(\\sec\\left(\\frac{π}{4}\\right)\\right)^{2}", "steps": [ { "type": "step", "result": "=\\sec^{2}\\left(\\frac{π}{4}\\right)" }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sec\\left(\\frac{π}{4}\\right)=\\sqrt{2}$$", "input": "\\sec\\left(\\frac{π}{4}\\right)", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\sqrt{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\left(\\sqrt{2}\\right)^{2}" }, { "type": "interim", "title": "Simplify $$\\left(\\sqrt{2}\\right)^{2}:{\\quad}2$$", "input": "\\left(\\sqrt{2}\\right)^{2}", "result": "=2", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\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": "=2^{\\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": "=2", "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": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nFuqJhqdbE8vRlM8Q9mP1HyRHuGw7+tM5METTDj6vVH6JwS9po8Lx246e07g7nTMZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz1xilETpud1E7a+EPTJfjQEialcV/dI5TH4fXyp+ncwuA==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }