Question

# Use ubtraction Formula to write the expression as a trigonometric function of one number.(cos)(13pi)/15cos(-pi/5)-(sin)(13pi)/15sin(-pi/5)

Trigonometric Functions
Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value.
$$\displaystyle{\left({\cos}\right)}\frac{{{13}\pi}}{{15}}{\cos{{\left(-\frac{\pi}{{5}}\right)}}}-{\left({\sin}\right)}\frac{{{13}\pi}}{{15}}{\sin{{\left(-\frac{\pi}{{5}}\right)}}}$$

2021-09-14
Our expression is
$$\displaystyle{\left({\cos}\right)}\frac{{{13}\pi}}{{15}}{\cos{{\left(-\frac{\pi}{{5}}\right)}}}-{\left({\sin}\right)}\frac{{{13}\pi}}{{15}}{\sin{{\left(-\frac{\pi}{{5}}\right)}}}$$
This kind of expression appears on the addition of angles formula for cosine
$$\displaystyle{\cos{{\left({x}+{y}\right)}}}={\cos{{x}}}{\cos{{y}}}-{\sin{{x}}}{\sin{{y}}}$$
with $$\displaystyle{x}=\frac{{{13}\pi}}{{15}},{y}=-\frac{\pi}{{5}}$$
So we have
$$\displaystyle{\left({\cos}\right)}\frac{{{13}\pi}}{{15}}{\cos{{\left(-\frac{\pi}{{5}}\right)}}}-{\left({\sin}\right)}\frac{{{13}\pi}}{{15}}{\sin{{\left(-\frac{\pi}{{5}}\right)}}}={\cos{{\left(\frac{{{13}\pi}}{{15}}+{\left(-\frac{\pi}{{5}}\right)}\right)}}}$$
$$\displaystyle={\cos{{\left(\frac{{{13}\pi}}{{15}}-\frac{{{3}\pi}}{{15}}\right)}}}$$
$$\displaystyle={\left({\cos}\right)}\frac{{{10}\pi}}{{15}}$$
$$\displaystyle={\left({\cos}\right)}\frac{{{2}\pi}}{{3}}$$
$$\displaystyle=-{\cos{{\left(\frac{\pi}{{3}}\right)}}}{\left[{\cos{{\left(\pi-{x}\right)}}}=-{\cos{{x}}}\right]}$$
$$\displaystyle=-\frac{{1}}{{2}}$$