Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/34

6. If $\scriptstyle {f(m_{1})={\frac {m_{1}-1}{m_{1}+1}}}$, show that

$\scriptstyle {{\frac {f(m_{1})-f(m_{2})}{1+f(m_{1})f(m_{2})}}={\frac {m_{1}-m_{2}}{1+m_{1}m_{2}}}}$.

7. If $\scriptstyle {\phi (x)=a^{x}}$, show that $\scriptstyle {\phi (y)\cdot \phi (z)=\phi (y+z)}$.

8. Given $\scriptstyle {\phi (x)=\log {\frac {1-x}{1+x}}}$; show that

$\scriptstyle {\phi (x)+\phi (y)=\phi \left({\frac {x+y}{1+xy}}\right)}$.

9. If $\scriptstyle {f(\phi )=\cos \phi }$, show that

$\scriptstyle {f(\phi )=f(-\phi )=-f(\pi -\phi )=-f(\pi +\phi )}$.

10. If $\scriptstyle {F(\theta )=\tan \theta }$, show that

$\scriptstyle {F(2\theta )={\frac {2F(\theta )}{1-[F(\theta )]^{2}}}}$.

11. Given $\scriptstyle {\psi (x)=x^{2n}+x^{2m}+1}$; show that

$\scriptstyle {\psi (1)=3,\quad \psi (0)=1,\quad \psi (a)=\psi (-a)}$.