how do I expand cos(n*theta)

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how do I expand cos(n*theta) into a polynomial in sin(theta) and cos(theta) for a range of integers n

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You could use the simplify_trig method:

sage: var('theta')
theta
sage: cos(2*theta).simplify_trig()
2*cos(x)^2 - 1
sage: [cos(n*theta).simplify_trig() for n in range(1, 10)]
...

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Last updated: Sep 13 '23