cos(15)cos(75)-sin(15)sin(75)

{ "query": { "display": "$$\\cos\\left(15^{\\circ\\:}\\right)\\cos\\left(75^{\\circ\\:}\\right)-\\sin\\left(15^{\\circ\\:}\\right)\\sin\\left(75^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\cos(15^{\\circ })\\cos(75^{\\circ })-\\sin(15^{\\circ })\\sin(75^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(15^{\\circ\\:}\\right)\\cos\\left(75^{\\circ\\:}\\right)-\\sin\\left(15^{\\circ\\:}\\right)\\sin\\left(75^{\\circ\\:}\\right)=0$$", "input": "\\cos\\left(15^{\\circ\\:}\\right)\\cos\\left(75^{\\circ\\:}\\right)-\\sin\\left(15^{\\circ\\:}\\right)\\sin\\left(75^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(90^{\\circ\\:}\\right)$$", "input": "\\cos\\left(15^{\\circ\\:}\\right)\\cos\\left(75^{\\circ\\:}\\right)-\\sin\\left(15^{\\circ\\:}\\right)\\sin\\left(75^{\\circ\\:}\\right)", "result": "=\\cos\\left(90^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Use the Angle Sum identity: $$\\cos\\left(s\\right)\\cos\\left(t\\right)-\\sin\\left(s\\right)\\sin\\left(t\\right)=\\cos\\left(s+t\\right)$$", "result": "=\\cos\\left(15^{\\circ\\:}+75^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\cos\\left(90^{\\circ\\:}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(90^{\\circ\\:}\\right)=0$$", "input": "\\cos\\left(90^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{2}&210^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{1}{2}&240^{\\circ }&-\\frac{1}{2}\\\\\\hline 90^{\\circ }&0&270^{\\circ }&0\\\\\\hline 120^{\\circ }&-\\frac{1}{2}&300^{\\circ }&\\frac{1}{2}\\\\\\hline 135^{\\circ }&-\\frac{\\sqrt{2}}{2}&315^{\\circ }&\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{2}&330^{\\circ }&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }