#6) Sum of 2 Angles
var('y')
show(sin(x+y))
show(sin(x+y).simplify_trig())
show(plot3d(sin(x+y),(x,-2,2),(y,-2,2)),viewer='tachyon')
show(plot3d(sin(x)*cos(y)+sin(y)*cos(x),(x,-2,2),(y,-2,2)))
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#7) Sum of 2 Angles
show(sin(x-y))
show(sin(x-y).simplify_trig())
show(plot3d(sin(x-y),(x,-2,2),(y,-2,2)),viewer='tachyon')
show(plot3d(sin(x)*cos(y)-sin(y)*cos(x),(x,-2,2),(y,-2,2)),viewer='tachyon')
 
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#8) Double Angle
show(cos(2*x))
show(cos(2*x).simplify_trig())
show(plot(cos(x)^2))
show(plot((1+cos(2*x))/2))
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#9) Double Angle
show(sin(2*x))
show(sin(2*x).simplify_trig())
show(plot(sin(x)^2))
show(plot((1-cos(2*x))/2))
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#10) Euler's Identity
show(taylor(e^x,x,0,8))
show(taylor(e^(i*x),x,0,8))
show(taylor(cos(x),x,0,8))
show(i*taylor(sin(x),x,0,8))
show(bool(e^(i*x)==cos(x)+i*sin(x)))
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#11) loose ends!
show(expand((x+1)^2))
show(factor(x^2+2*x+1))
show(diff(x^2+2*x+1,x))
show(diff((x+1)^2,x))
show(integrate(x^2+2*x+1,x,0,1))
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