To prove $\tan \phi_m + \sec \phi_m =(\tan \phi_1 + \sec \phi_1)^m $

Question

If $\phi_1, \phi_2, ... $ is a series of positive acute angles so that $\tan \phi_{m+1} = \tan \phi_m \sec \phi_1 + \sec \phi_m \tan \phi_1$ then prove that- $$\tan \phi_{m+n} = \tan \phi_m \sec \phi_n + \sec \phi_m \tan \phi_n$$ $$\sec \phi_{m+n} = \sec \phi_m \sec \phi_n + \tan \phi_m \tan \phi_n$$ $$\tan \phi_m + \sec \phi_m =(\tan \phi_1 + \sec \phi_1)^m $$

Answer 1

Let $t:=\text{arctanh}\big(\sin(\phi_1)\big)\in\mathbb{R}_{>0}$. Observe that $$\sin(\phi_1)=\tanh(t)\,,\,\,\cos(\phi_1)=\text{sech}(t)\,,\,\,\tan(\phi_1)=\sinh(t)\,,\text{ and }\sec(\phi_1)=\cosh(t)\,.$$ It follows that $$\sin(\phi_2)=\tanh(2t)\,,\,\,\cos(\phi_2)=\text{sech}(2t)\,,\,\,\tan(\phi_2)=\sinh(2t)\,,\text{ and }\sec(\phi_2)=\cosh(2t)\,.$$ By induction, $$\sin(\phi_m)=\tanh(mt)\,,\,\,\cos(\phi_m)=\text{sech}(mt)\,,\,\,\tan(\phi_m)=\sinh(mt)\,,\text{ and }\sec(\phi_m)=\cosh(mt)\,,$$ for all $m=1,2,3,\ldots$. That is, $$\begin{align}\tan(\phi_{m+n})&=\sinh\big((m+n)t\big)\\&=\sinh(mt)\,\cosh(nt)+\cosh(mt)\,\sinh(nt)\\&=\tan(\phi_m)\,\sec(\phi_n)+\sec(\phi_m)\,\tan(\phi_n)\,,\end{align}$$ for each $m,n=1,2,3,\ldots$. Similarly, $$\begin{align}\sec(\phi_{m+n})&=\cosh\big((m+n)t\big)\\&=\cosh(mt)\,\cosh(nt)+\sinh(mt)\,\sinh(nt)\\&=\sec(\phi_m)\,\sec(\phi_n)+\tan(\phi_m)\,\tan(\phi_n)\,,\end{align}$$ for each $m,n=1,2,3,\ldots$. Finally, $$\begin{align}\tan(\phi_m)+\sec(\phi_m)&=\sinh(mt)+\cosh(mt)=\exp(mt)=\big(\exp(t)\big)^m\\&=\big(\sinh(t)+\cosh(t)\big)^m=\big(\tan(\phi_1)+\sec(\phi_1)\big)^m \end{align}$$ for every $m=1,2,3,\ldots$.

Answer 2

To prove 1st statement, there is nothing to be do, it is given in the question,

for second statement write $sec\theta =\sqrt{1+tan^2\theta }$ and then write $1=tan^2\theta + sec^2\theta$