Question

Show that $\cos hx+\sin hx=e^x$ and simplify $\cos hx-\sin hx=?.$By considering ${(\cos hx+\sin hx)}^2+{(\cos hx-\sin hx)}^2$ show that ${\cos }^{2}hx-{\sin }^{2}hx=\cos h2x$.

Answer 1

We know that;

$coshx=\frac{e^x+e^{-x}}{2}$

$sinhx=\frac{e^x-e^{-x}}{2}$

Thus;

$coshx+sinhx=\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=e^x$

$coshx-sinhx=\frac{e^x+e^{-x}}{2}-\frac{e^x-e^{-x}}{2}=e^{-x}$

Thus;

$(coshx+sinhx{)}^2=e^{2x}$

$(coshx-sinhx{)}^2=e^{-2x}$

$\to$$(coshx+sinhx{)}^2+$$(coshx-sinhx{)}^2=e^{2x}+e^{-2x}$

$\to 2(cos{h}^{2}hx+si{n}^{2}hx)=e^{2x}+e^{-2x}$

$\to (cos{h}^{2}hx+si{n}^{2}hx)=\frac{e^{2x}+e^{-2x}}{2}=cosh2x$