Newton's 1676 formulas are $$ {}_2F_1\left(\genfrac{}{}{0pt}{0}{{1}/{2}+a,{1}/{2}-a}{{1}/{2}};\sin^2x\right)\cos x=\cos(2ax),\tag{1} $$ $$ {}_2F_1\left(\genfrac{}{}{0pt}{0}{a,-a}{{1}/{2}};\sin^2x\right)=\cos(2ax),\tag{2} $$ $$ {}_2F_1\left(\genfrac{}{}{0pt}{0}{{1}/{2}+a,{1}/{2}-a}{{3}/{2}};\sin^2x\right)\sin x=\frac{\sin(2ax)}{2a},\tag{3} $$
15.4.12, 15.4.14, 15.4.16 in https://dlmf.nist.gov/15.4
The following formulas are generalizations of the formulas above:
$$ {}_3F_2\left(\genfrac{}{}{0pt}{0}{1,a,b}{(a+b)/2,{(1+a+b)}/{2}};\sin^2x\right)\cos x\\ =\frac{a+b-1}{b-1}\sum_{n=0}^\infty \frac{(a)_n}{(2-b)_n}\cos(2n+1)x+\frac{\Gamma(a+b)\Gamma(1-b)}{\Gamma(a)(2\sin x)^{a+b-1}}\sin\left(\tfrac{\pi}{2}(a+b)+x(b-a)\right),\tag{4} $$
$$ \frac{\Gamma(1-a)\Gamma(1-b)}{2\Gamma(1-a-b)}\,{}_2F_1\left(\genfrac{}{}{0pt}{0}{a,b}{{1}/{2}};\cos^2x\right)=\frac{1}{2}+\sum_{n=1}^\infty\frac{(a)_n(b)_n}{(1-a)_n(1-b)_n}\cos(2nx),\tag{5} $$
$$ \frac{\Gamma(2-a)\Gamma(2-b)}{2\Gamma(2-a-b)}\,{}_2F_1\left(\genfrac{}{}{0pt}{0}{a,b}{{3}/{2}};\cos^2x\right)\cos x=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(2-a)_n(2-b)_n}\cos(2n+1)x.\tag{6} $$
For example, when $a+b=1$, (4) reduces to (1).
When $b=−a$, (5) reduces to (2), because RHS of (5) in this case is a Fourier series expansion of $\cos(2ax)$. Similarly for (6) and (3).
Q: Are any other generalizations of Newton's formulas known?