## review derivation: Euler equation: trig square root

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
16 declare final expr
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
no validation is available for declarations 9988949211:
9988949211:
8 declare identity
1. 5832984291; locally 9385720:
$$(\sin(x))^2 + (\cos(x))^2 = 1$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)} = 1$$
no validation is available for declarations 5832984291: error for dim with 5832984291
5832984291: N/A
15 divide both sides by
1. 9889984281; locally 7472666:
$$2 (\sin(x))^2 = 1 - \cos(2 x)$$
$$2 \sin^{2}{\left(pdg_{1464} \right)} = 1 - \cos{\left(2 pdg_{1464} \right)}$$
1. 0003838111:
$$2$$
$$2$$
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
valid 9889984281:
9988949211:
9889984281:
9988949211:
10 swap LHS with RHS
1. 9482928243; locally 4890284:
$$\cos(2 x) + (\sin(x))^2 = (\cos(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = \cos^{2}{\left(pdg_{1464} \right)}$$
1. 9482438243; locally 2936550:
$$(\cos(x))^2 = \cos(2 x) + (\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = \sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
valid 9482928243:
9482438243:
9482928243:
9482438243:
5 expand RHS
1. 4638429483; locally 3333333:
$$\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))$$
$$e^{2 pdg_{1464} pdg_{4621}} = \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)^{2}$$
1. 4598294821; locally 4444444:
$$\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2$$
$$e^{2 pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
LHS diff is 0 RHS diff is (pdg4621**2 + 1)*sin(pdg1464)**2 4638429483:
4598294821:
4638429483:
4598294821:
2 change variable X to Y
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0004089571:
$$2 x$$
$$2 pdg_{1464}$$
2. 0004582412:
$$x$$
$$pdg_{1464}$$
1. 4838429483; locally 9999999:
$$\exp(2 i x) = \cos(2 x)+i \sin(2 x)$$
$$e^{2 pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
LHS diff is (1 - exp(pdg1464*pdg4621))*exp(pdg1464*pdg4621) RHS diff is pdg4621*sin(pdg1464) - pdg4621*sin(2*pdg1464) + cos(pdg1464) - cos(2*pdg1464) 4938429483:
4838429483:
4938429483:
4838429483:
12 subtract X from both sides
1. 4827492911; locally 9481000:
$$\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
1. 0006466214:
$$(\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)}$$
1. 1248277773; locally 7472641:
$$\cos(2 x) = 1 - 2 (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = 1 - 2 \sin^{2}{\left(pdg_{1464} \right)}$$
valid 4827492911:
1248277773:
4827492911:
1248277773:
14 subtract X from both sides
1. 7572664728; locally 1029911:
$$\cos(2 x) + 2 (\sin(x))^2 = 1$$
$$2 \sin^{2}{\left(pdg_{4037} \right)} + \cos{\left(2 pdg_{4037} \right)} = 1$$
1. 0008842811:
$$\cos(2 x)$$
$$\cos{\left(2 pdg_{1464} \right)}$$
1. 9889984281; locally 7472666:
$$2 (\sin(x))^2 = 1 - \cos(2 x)$$
$$2 \sin^{2}{\left(pdg_{1464} \right)} = 1 - \cos{\left(2 pdg_{1464} \right)}$$
valid 7572664728:
9889984281:
7572664728:
9889984281:
13 add X to both sides
1. 1248277773; locally 7472641:
$$\cos(2 x) = 1 - 2 (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = 1 - 2 \sin^{2}{\left(pdg_{1464} \right)}$$
1. 0007471778:
$$2(\sin(x))^2$$
$$2 \sin^{2}{\left(pdg_{1464} \right)}$$
1. 7572664728; locally 1029911:
$$\cos(2 x) + 2 (\sin(x))^2 = 1$$
$$2 \sin^{2}{\left(pdg_{4037} \right)} + \cos{\left(2 pdg_{4037} \right)} = 1$$
valid 1248277773:
7572664728:
1248277773:
7572664728:
1 declare initial expr
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 4938429483:
4938429483:
6 select real parts
1. 9483928192; locally 2222222:
$$\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2$$
$$pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
1. 9482928242; locally 5828294:
$$\cos(2 x) = (\cos(x))^2 - (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
LHS diff is -cos(2*pdg1464) + cos(2*re(pdg1464))*cosh(2*im(pdg1464)) + re(pdg4621*sin(2*pdg1464)) RHS diff is -cos(2*pdg1464) + 2*cos(2*re(pdg1464))*sinh(im(pdg1464))**2 + cos(2*re(pdg1464)) + re(pdg4621*sin(2*pdg1464)) 9483928192:
9482928242:
9483928192:
9482928242:
7 add X to both sides
1. 9482928242; locally 5828294:
$$\cos(2 x) = (\cos(x))^2 - (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
1. 0007894942:
$$(\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)}$$
1. 9482928243; locally 4890284:
$$\cos(2 x) + (\sin(x))^2 = (\cos(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = \cos^{2}{\left(pdg_{1464} \right)}$$
valid 9482928242:
9482928243:
9482928242:
9482928243:
4 multiply expr 1 by expr 2
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
2. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 4638429483; locally 3333333:
$$\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))$$
$$e^{2 pdg_{1464} pdg_{4621}} = \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)^{2}$$
valid 4938429483:
4938429483:
4638429483:
4938429483:
4938429483:
4638429483:
11 LHS of expr 1 equals LHS of expr 2
1. 9482438243; locally 2936550:
$$(\cos(x))^2 = \cos(2 x) + (\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = \sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
2. 3285732911; locally 9123670:
$$(\cos(x))^2 = 1-(\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
1. 4827492911; locally 9481000:
$$\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
valid 9482438243:
3285732911: error for dim with 3285732911
4827492911:
9482438243:
3285732911: N/A
4827492911:
3 LHS of expr 1 equals LHS of expr 2
1. 4598294821; locally 4444444:
$$\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2$$
$$e^{2 pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
2. 4838429483; locally 9999999:
$$\exp(2 i x) = \cos(2 x)+i \sin(2 x)$$
$$e^{2 pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
1. 9483928192; locally 2222222:
$$\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2$$
$$pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
valid 4598294821:
4838429483:
9483928192:
4598294821:
4838429483:
9483928192:
9 subtract X from both sides
1. 5832984291; locally 9385720:
$$(\sin(x))^2 + (\cos(x))^2 = 1$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)} = 1$$
1. 0001111111:
$$(\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)}$$
1. 3285732911; locally 9123670:
$$(\cos(x))^2 = 1-(\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
valid 5832984291: error for dim with 5832984291
3285732911: error for dim with 3285732911
5832984291: N/A
3285732911: N/A
Physics Derivation Graph: Steps for Euler equation: trig square root

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
4037 variable x
$$x$$
['real']
• length: 1
position
53
4621 variable i
$$i$$
['imaginary'] dimensionless imaginary unit
74
1464 variable x
$$x$$
['real'] dimensionless 140
MESSAGE:
• local variable 'all_df' referenced before assignment