How to do this sagemath task in Jupyter about Ring of Symbolic Expressions – SR

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1) create a holded symbolic expression of the form z = cos( pi/2 )^2/ sin(pi)^3

2) take z from last subtask and substitute the function ‘cos’ by ‘sin’ and unhold the result.

3) let v be the Expression v = (x+y)^2 +(y+z)^2 +(x+z)^2. Use the substitution and only subsitution to turn v into x^2

4) let v like in last subtask. Find a single substitution for x, y and z such that v becomes the x^2 +y^2 +z^2.

5) First assume x to be real. Use sage to show, that ALPHA((cos(x) + i sin(x))^4) = cos(4x).

6) Proof by root test, that ∑∞ to k=0 = k^3/2^k is convergent. Compute ∑∞ to k=0 = k^3/2^k.

7) compute the hessian and jacobian of f : R^3 → R, f(x,y,z) = y · sin(2x − y^4 + e^z).