Jerold

2021-08-15

Use an Addition or Subpraction Formula to write the expression as a trigonometric function of one number:
$\frac{{\mathrm{tan}78}^{\circ }-{\mathrm{tan}18}^{\circ }}{1+{\mathrm{tan}78}^{\circ }{\mathrm{tan}18}^{\circ }}$
Find its exact value.

oppturf

The given trigonometry expression can be expressed as difference formula of tan angle which is given as:
$\mathrm{tan}\left(A-B\right)=\frac{\mathrm{tan}A-\mathrm{tan}B}{1+\mathrm{tan}A\mathrm{tan}B}$
where A and B are in degree.
Comparing with the given expression, we get:
$A={78}^{\circ }$ and $B={18}^{\circ }$
Put these value in the tan angle formula as:
$\mathrm{tan}\left({78}^{\circ }-{18}^{\circ }\right)=\frac{{\mathrm{tan}78}^{\circ }-{\mathrm{tan}18}^{\circ }}{1+{\mathrm{tan}78}^{\circ }{\mathrm{tan}18}^{\circ }}$
$\mathrm{tan}\left({60}^{\circ }\right)=\sqrt{3}$
The exact value of given expression can be calculated as:
$\frac{{\mathrm{tan}78}^{\circ }-{\mathrm{tan}18}^{\circ }}{1+{\mathrm{tan}78}^{\circ }{\mathrm{tan}18}^{\circ }}=\frac{4.7046-0.3249}{2.5286}$
$=\frac{4.3797}{2.5286}$
$=1.73206$
Therefore, the exact value of given expression is 1.73206.

Jeffrey Jordon