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WorksForMe(TM):

sage: var("t")
t
sage: theta=function("theta")
sage: A=cos(theta(t))
sage: diff(A,t)
-sin(theta(t))*diff(theta(t), t)
sage: var("B,C")
(B, C)
sage: diff(A,t).subs([diff(theta(t),t)==B,theta(t)==C])
-B*sin(C)

But remember that in the subs method, the single equal sign (which is

WorksForMe(TM):

sage: var("t")
t
sage: theta=function("theta")
sage: A=cos(theta(t))
sage: diff(A,t)
-sin(theta(t))*diff(theta(t), t)
sage: var("B,C")
(B, C)
sage: diff(A,t).subs([diff(theta(t),t)==B,theta(t)==C])
-B*sin(C)

But remember that in the subs method, the single equal sign (which is a marker for keywords in a function call) is accepted for a single substitition. To to multiple substitutions, use either a list of equations (whose sides are separated by ==, i. e. a double equal sign) or a dictionary.