Let $n$, $m$ and $k$ be some (positive) integers such that $(k+3/2)-(n+m/2)<0$. Can the hypergeometric function $$F\left (n+\frac{m}{2},n+\frac{m+1}{2};k+\frac{3}{2};-\tan^2{\phi}\right) \tag{1}$$ be expressed through an expression which is convenient for numerical computation of its value? For example, when $k=0$ we have $$F\left (n+\frac{m}{2},n+\frac{m+1}{2};\frac{3}{2};-\tan^2{\phi}\right)=(\cos{\phi})^{2n+m}\frac{\sin{[(2n+m-1)\phi]}}{(2n+m-1)\sin{\phi}}.$$

**P.S.** Using the first two transformation formulas indicated by @Johannes Trost, we can get
$$F\left (n+\frac{m}{2},n+\frac{m+1}{2};k+\frac{3}{2};-\tan^2{\phi}\right)=[\cos{\phi}]^{2(n+M)}\frac{P_{n+M-k-1}^{(k+1/2,-1/2)}(\cos{(2\phi)}}{P_{n+M-k-1}^{(k+1/2,-1/2)}(1)}$$ if $m=2M$, and
$$F\left (n+\frac{m}{2},n+\frac{m+1}{2};k+\frac{3}{2};-\tan^2{\phi}\right)=[\cos{\phi}]^{2(n+M+1)}\frac{P_{n+M-k-1}^{(k+1/2,1/2)}(\cos{(2\phi)}}{P_{n+M-k-1}^{(k+1/2,1/2)}(1)}$$ if $m=2M+1$. Here $P_n^{(\alpha,\beta)}$ are Jacobi polynomials.

These can be considered as a generalization of the above given formula for $k=0$ because $$P_n^{(1/2,1/2)}(\cos{2\phi})=2\frac{(2n+1)!!}{(2n+2)!!}\frac{\sin{[2(n+1)\phi]}}{\sin{2\phi}},$$ and $$P_n^{(1/2,-1/2)}(\cos{2\phi})=\frac{(2n-1)!!}{(2n)!!}\frac{\sin{[(2n+1)\phi]}}{\sin{\phi}}.$$