prove sin(x)cos(y)= 1/2 (sin(x+y)+sin(x-y))

{ "query": { "display": "prove $$\\sin\\left(x\\right)\\cos\\left(y\\right)=\\frac{1}{2}\\left(\\sin\\left(x+y\\right)+\\sin\\left(x-y\\right)\\right)$$", "symbolab_question": "TRIG_PROVING#prove \\sin(x)\\cos(y)=\\frac{1}{2}(\\sin(x+y)+\\sin(x-y))" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Identities", "subTopic": "Other", "default": "\\mathrm{True}" }, "steps": { "type": "interim", "title": "Prove $$\\sin\\left(x\\right)\\cos\\left(y\\right)=\\frac{1}{2}\\left(\\sin\\left(x+y\\right)+\\sin\\left(x-y\\right)\\right):{\\quad}$$True", "input": "\\sin\\left(x\\right)\\cos\\left(y\\right)=\\frac{1}{2}\\left(\\sin\\left(x+y\\right)+\\sin\\left(x-y\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the Product to Sum identity: $$\\sin\\left(s\\right)\\cos\\left(t\\right)=\\frac{1}{2}\\left(\\sin\\left(s+t\\right)+\\sin\\left(s-t\\right)\\right)$$" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "solvingClass": "Trig Proving", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Identities", "practiceTopic": "Trig Identities" } } }