prove cos(2θ)=2cos^2(θ)-1

{ "query": { "display": "prove $$\\cos\\left(2θ\\right)=2\\cos^{2}\\left(θ\\right)-1$$", "symbolab_question": "TRIG_PROVING#prove \\cos(2θ)=2\\cos^{2}(θ)-1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Identities", "subTopic": "Other", "default": "\\mathrm{True}" }, "steps": { "type": "interim", "title": "Prove $$\\cos\\left(2θ\\right)=2\\cos^{2}\\left(θ\\right)-1:{\\quad}$$True", "input": "\\cos\\left(2θ\\right)=2\\cos^{2}\\left(θ\\right)-1", "steps": [ { "type": "step", "primary": "Manipulating left side", "secondary": [ "$$\\cos\\left(2θ\\right)$$" ] }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(2θ\\right)", "result": "=\\cos^{2}\\left(θ\\right)-\\sin^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=\\cos\\left(θ+θ\\right)" }, { "type": "step", "primary": "Use the Angle Sum identity: $$\\cos\\left(s+s\\right)=\\cos\\left(s\\right)\\cos\\left(s\\right)-\\sin\\left(s\\right)\\sin\\left(s\\right)$$", "result": "=\\cos\\left(θ\\right)\\cos\\left(θ\\right)-\\sin\\left(θ\\right)\\sin\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(θ\\right)\\cos\\left(θ\\right)-\\sin\\left(θ\\right)\\sin\\left(θ\\right):{\\quad}\\cos^{2}\\left(θ\\right)-\\sin^{2}\\left(θ\\right)$$", "input": "\\cos\\left(θ\\right)\\cos\\left(θ\\right)-\\sin\\left(θ\\right)\\sin\\left(θ\\right)", "result": "=\\cos^{2}\\left(θ\\right)-\\sin^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(θ\\right)\\cos\\left(θ\\right)=\\cos^{2}\\left(θ\\right)$$", "input": "\\cos\\left(θ\\right)\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\cos\\left(θ\\right)\\cos\\left(θ\\right)=\\:\\cos^{1+1}\\left(θ\\right)$$" ], "result": "=\\cos^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\cos^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X+Ns6q7JPRMfm0PI0xvLqTHt7fABx6LEy6vdX9TQoXDMwViaLUXkeD+JukROhWdjgptkicZFer8E7JPqjDOf+aN6Hv6MoTMtvtU0IQwXdn8lX0liVxNuZc7FxloDSGMHTgesDAWxNAdGPgtb21qRBiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)\\sin\\left(θ\\right)=\\sin^{2}\\left(θ\\right)$$", "input": "\\sin\\left(θ\\right)\\sin\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sin\\left(θ\\right)\\sin\\left(θ\\right)=\\:\\sin^{1+1}\\left(θ\\right)$$" ], "result": "=\\sin^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\sin^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ucwh1lTcOkNeom/kNAu3YTHt7fABx6LEy6vdX9TQoXDMwViaLUXkeD+JukROhWdjVWq9IfcTz6XeNlITmlpBcKN6Hv6MoTMtvtU0IQwXdn/aPPXxzbPJbRss6a7Z6DSgVMJ2Q4xq+XDOIRYIVDhneiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\cos^{2}\\left(θ\\right)-\\sin^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbHF7IXqH4A4QSliMFYTTdaQSwtk/DRWb0wPqX3+QHVz3V/ZUTeJpWKv8yNHYMRFxnT73Z4YtLN5Wfgye6xIk3qmdEmAakeuFuYhE8idCkHNwVkV6lnN+hX/6ZwXFGc3ao6N6Hv6MoTMtvtU0IQwXdn8BSh0OkRlUI4ljquR9dxWSpo5Agsp/TndH0MglppaTsxzq1+1M1jYqxS29VuKznM0=" } }, { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(θ\\right)+\\sin^{2}\\left(θ\\right)=1$$", "secondary": [ "$$\\sin^{2}\\left(θ\\right)=1-\\cos^{2}\\left(θ\\right)$$" ], "result": "=\\cos^{2}\\left(θ\\right)-\\left(1-\\cos^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\cos^{2}\\left(θ\\right)-\\left(1-\\cos^{2}\\left(θ\\right)\\right):{\\quad}2\\cos^{2}\\left(θ\\right)-1$$", "input": "\\cos^{2}\\left(θ\\right)-\\left(1-\\cos^{2}\\left(θ\\right)\\right)", "result": "=2\\cos^{2}\\left(θ\\right)-1", "steps": [ { "type": "interim", "title": "$$-\\left(1-\\cos^{2}\\left(θ\\right)\\right):{\\quad}-1+\\cos^{2}\\left(θ\\right)$$", "input": "-\\left(1-\\cos^{2}\\left(θ\\right)\\right)", "result": "=\\cos^{2}\\left(θ\\right)-1+\\cos^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-1-\\left(-\\cos^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-1+\\cos^{2}\\left(θ\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Simplify $$\\cos^{2}\\left(θ\\right)-1+\\cos^{2}\\left(θ\\right):{\\quad}2\\cos^{2}\\left(θ\\right)-1$$", "input": "\\cos^{2}\\left(θ\\right)-1+\\cos^{2}\\left(θ\\right)", "result": "=2\\cos^{2}\\left(θ\\right)-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\cos^{2}\\left(θ\\right)+\\cos^{2}\\left(θ\\right)-1" }, { "type": "step", "primary": "Add similar elements: $$\\cos^{2}\\left(θ\\right)+\\cos^{2}\\left(θ\\right)=2\\cos^{2}\\left(θ\\right)$$", "result": "=2\\cos^{2}\\left(θ\\right)-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "We showed that the two sides could take the same form", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "solvingClass": "Trig Proving", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Identities", "practiceTopic": "Trig Identities" } } }